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2.1: Functions, Limits, and Continuity

Difficulty Level: At Grade Created by: CK-12

This Calculus Common Errors accompanies the CK-12 Foundation's Calculus Student Edition.

Lesson 1: Equations and Graphs

To begin the study of calculus, it is helpful to review some important properties of equations and functions, and how to graph different kinds of functions on an \begin{align*}x-y\end{align*} coordinate system. A solid understanding of analytic geometry is essential to developing the techniques of differentiation and integration presented in this textbook. The ability to identify analytic solutions to the points where graphs intersect the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} axes (e.g. the intercepts), as well as finding the exact points where two graphs or curves cross each other, will be necessary to evaluate limits, derivatives and integrals.

An important technique students will need throughout the study of calculus is evaluating functions by substituting in a value for a function's argument. In simple cases, the argument is given as a number. Given a function \begin{align*}f(x) = x^2\end{align*}, to find \begin{align*}f(4)\end{align*} we substitute the \begin{align*}4\end{align*} in for \begin{align*}x\end{align*}, and get \begin{align*}f(4) = 4^2 = 16\end{align*}.

Students must soon become comfortable with substituting entire algebraic expressions in for the argument of a function and evaluating the output. For instance, when calculating the derivative of a function, students will need to evaluate expressions like \begin{align*}f(x+a)\end{align*} for a variety of functions. For instance, if \begin{align*}f(x) = x^2 + 2x + 3\end{align*}, to calculate \begin{align*}f(x+a)\end{align*} we must substitute \begin{align*}x+a\end{align*} for the value of \begin{align*}x\end{align*} in the original function. This gives us:

\begin{align*}f(x) & = (x+a)^2 + 2(x+a) + 3 \\ f(x+a) & = (x+a)(x+a) + 2x+2a+3 \\ f(x+a) & = x^2 + 2ax + a^2 + 2x + 3\end{align*}

Because polynomials are usually grouped into like terms, and the letter \begin{align*}``a"\end{align*} in this case is a constant (i.e. not a variable), we would rewrite this expression as:

\begin{align*}f(x+a) = x^2 + (2a+a^2)x+a^2 + 3\end{align*}

In this process, we have done nothing more than apply the rules of algebra to our function, but the process of evaluating functions with algebraic expressions as arguments will be unfamiliar to many students. Many will try to use some sort of shortcut to avoid expanding terms as necessary.

An important distinction should be drawn between the terms “function” and “equation”, and how the graphical representation of a function can help us to solve an equation. For example, the table on page \begin{align*}1\end{align*} displays the output values for \begin{align*}f(x) = x^2\end{align*} when evaluated for different values of the input variable \begin{align*}x\end{align*}. This enables us to graph the function on the \begin{align*}x-y\end{align*} plane for any value of \begin{align*}x\end{align*}, as seen on the top of page 2.

Alternatively, when we consider an equation with \begin{align*}x^2\end{align*} in it, for instance \begin{align*}x^2 = 4\end{align*}, we are asking for the specific point on the graph of \begin{align*}f(x) = x^2\end{align*} that equals \begin{align*}4\end{align*}. Instead of the expression containing the dependent variable \begin{align*}y\end{align*} (or \begin{align*}f(x)\end{align*}), we are substituting a specific numerical value for the dependent variable, and determining what value or values of the independent variable \begin{align*}x\end{align*} that satisfy this condition. Whereas a function represents the general rule to calculate the value of \begin{align*}f(x)\end{align*} for any input value, an equation asks for the value or values of the input that yields a specific output value for \begin{align*}y\end{align*}. It therefore usually only has a finite set of answers. On a graph, an equation corresponds to particular points on the curve we have drawn, whereas a function refers to the entire curve.

In the case of \begin{align*}x^2 = 4\end{align*}, there are two points on the parabola where \begin{align*}f(x)\end{align*} or \begin{align*}y\end{align*} equals \begin{align*}4\end{align*}, so there exist two answers to this equation: \begin{align*}x = +2\end{align*}, and \begin{align*}x = -2\end{align*}.

Similarly, when calculating the \begin{align*}y-\end{align*}intercept of a function, such as \begin{align*}y = 2x + 3\end{align*}, we are asking for the \begin{align*}y-\end{align*}value when \begin{align*}x = 0\end{align*}. So we would substitute the value \begin{align*}0\end{align*} for \begin{align*}x\end{align*}, and arrive at the equation

\begin{align*}2(0) + 3 = y\end{align*}

which tells us the value of \begin{align*}y\end{align*} when \begin{align*}x\end{align*} equals \begin{align*}0\end{align*}, called the y-intercept. In this case, the \begin{align*}y-\end{align*}intercept is \begin{align*}3\end{align*}.

The graph on page 3 illustrates the relationships between graphs and equations, by setting the values of two functions equal to each other. If one function is represented by \begin{align*}f(x) = 2x + 3\end{align*}, and the other is represented by \begin{align*}g(x) = x^2 + 2x - 1\end{align*}, to find the points where the graphs of these two curves intersect entails finding the place where \begin{align*}f(x) = g(x)\end{align*} for a given value of \begin{align*}x\end{align*}. To determine these values, we write the equation

\begin{align*}2x+3=x^2+2x-1\end{align*}

and solve for the values of \begin{align*}x\end{align*} where this equation is true. This example requires using techniques for solving quadratic equations, as shown on Page 3. Again it turns out that there are two answers for \begin{align*}x\end{align*}, corresponding to the two points of intersection for the graphs of the functions \begin{align*}f(x)\end{align*} and \begin{align*}g(x)\end{align*}.

Although much of the notation introduced in this and subsequent lessons is very formal, it is important to stress that functions are important because they enable us to model a number of real world phenomena. In the exercises for this chapter, the relationship between the independent variable \begin{align*}x\end{align*}, and the example of modeling costs using both linear and nonlinear functions, is emphasized. By using functions to model real world phenomena, we find that properties of functions like slopes and intercepts correspond to actual real world phenomena, like break even points, fixed and variable costs, as well as velocity and acceleration.

Lesson 2: Relations and Functions

In Lesson 2, the more formal definition of a function is introduced, as are the topics of Domain and Range which provide useful information for analyzing and graphing functions. The classic definition of a function when displayed graphically is that it is a curve on the \begin{align*}x-y\end{align*} plane that must satisfy the “vertical line test”, e.g. if you draw a vertical line through the function, it touches the graph in at most one point. This test ensures that for any value of \begin{align*}x\end{align*}, there is at most only one value that our function evaluates to for this input. This is sometimes referred to as being “onto” or “surjective”.

The example given in this Lesson of a common graphical representation which is NOT a function is the graph of a circle. There are clearly places where if we were to draw a vertical line on the coordinate plane, it would cross the circle twice. Although the rationale behind this isn't explained in this Lesson, it may be helpful for students to be shown why an equation like \begin{align*}x^2+y^2 = 4\end{align*} will not be a function, whereas an equation like \begin{align*}x^2 + y = 4\end{align*} does turn out to be a function. The answer becomes clear if we were to isolate the variable \begin{align*}y\end{align*}:

\begin{align*}x^2 + y^2 & = 4 \\ y^2 & = 4 - x^2 \\ y & = \pm \sqrt{(4 - x^2)}\end{align*}

By isolating y, we see that for a given input value of \begin{align*}x\end{align*}, there can be two values of y due to the plus/minus in the square root. Because there are two output values for only one input value, this is NOT a function, and thus does not pass the vertical line test.

When treating functions in the context of the \begin{align*}x-y\end{align*} plane, it often appears that the variety of curves that are functions is very limited, since there are a number of interesting curves which don't satisfy the vertical line test. These include the circle graphed in the text, and the spiral graphed below. Can the techniques we develop to analyze functions be applied to these non-functions?

Although it is outside the scope of this textbook, most students will have been introduced to the concept of a “parametrized curve” in their pre-calculus course. Parametrizing a curve enables us to consider curves that are not functions, like the circle or spiral, and represent them AS functions so that we can analyze them using function-based techniques. This entails creating a new variable, or parameter, and re-writing our expressions for the \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}coordinates of our curve using this parameter. For instance, if we were to create a new variable named t, referred to as our parameter, we could describe the circle in this Lesson using the equations:

\begin{align*}x = \cos (t), y = \sin(t), 0 \le t \le 2\pi\end{align*}

In this case, both of our “parameterized” equations ARE functions: \begin{align*}\cos(t)\end{align*} and \begin{align*}\sin(t)\end{align*}. By using techniques like parameterization, we can transform curves that are not functions into representations which ARE functions. This dramatically increases the class of curves and graphs which we can analyze.

The bulk of this lesson is devoted to reviewing the topic of a function's Domain and Range, which define the values of \begin{align*}x\end{align*} and \begin{align*}y\end{align*} over which a given function extends.

Determining the domain of a function is usually much easier for students than finding its range, since there are only a finite number of situations where we cannot evaluate a function at a given \begin{align*}x-\end{align*}value. The two most common are dividing by zero and taking the square root of a negative number. In looking at a function to determine its domain, most often we are simply looking for cases where a particular value of \begin{align*}x\end{align*} will lead us into one of these conditions of undefinedness, and exclude those values.

Take, for instance, the example of the rational function \begin{align*}f(x) = \frac{1}{x}\end{align*}. In determining the values of \begin{align*}x\end{align*} for which this function exists, clearly we must exclude the value \begin{align*}x = 0\end{align*} since we are not allowed to divide by zero. Since there are no other opportunities for our equation to be undefined through either dividing by zero or taking the square root of a negative number, this is the only point excluded in our domain. We can therefore define the domain as:

\begin{align*} D = \left \{x | x \neq 0 \right \} \end{align*}

Similarly, if we were to look at the rational function:

\begin{align*}f(x) = \frac{1}{(x-2)(x+3)}\end{align*}

the denominator in this expression will be equal to zero when the product \begin{align*}(x-2)(x+3) = 0\end{align*}. This happens when \begin{align*}x = 2\end{align*} or \begin{align*}x = -3\end{align*}. In this case, our domain includes all values of \begin{align*}x\end{align*} except for \begin{align*}x = 2\end{align*} and \begin{align*}x = -3\end{align*}.

The determination of a function's range is much more complicated, since it often requires a great deal of intuition into the behavior of algebraic expressions to understand which values a complicated function can and cannot take. For instance, terms in polynomials which raise \begin{align*}x\end{align*} to an even power will always be positive, and the sine or cosine of a variable will always range between \begin{align*}-1\end{align*} and \begin{align*}1\end{align*}. An excellent process to help students identify the range of a function, particularly one that has many terms, is to look at the range of the individual terms, and combine them through logical reasoning to determine the range of the entire functions.

For example, consider the following function:

\begin{align*}f(x) = x^2 + \cos(x)\end{align*}

Can we determine what values \begin{align*}f(x)\end{align*} will take by just looking at this expression? Looking at the first term, we know that the range of \begin{align*}x^2\end{align*} is always greater than or equal \begin{align*}0\end{align*}, since \begin{align*}x^2\end{align*} can never be negative. Moving to the next term, \begin{align*}\cos(x)\end{align*}, which we know cosine is always between \begin{align*}+1\end{align*} and \begin{align*}-1\end{align*}. Combining these facts, we see that \begin{align*}f(x)\end{align*} can never get less than \begin{align*}-1\end{align*}, but can grow positively as large as we want due to the term, \begin{align*}x^2\end{align*}. We can therefore say that the Range of this function is at the very least \begin{align*}f(x) > -1\end{align*}, since \begin{align*}f(x)\end{align*} can only get as small as \begin{align*}-1\end{align*}. Though it turns out that the range is actually more restricted than this, this type of reasoning provides students with an example of bounding a range to a particular interval.

To many students, understanding the domain and range often becomes formulaic, with little or no motivation as to why these terms are important. In keeping with the importance of understanding both the practical applications of functions, and being able to graph functions and identify functions from their graphs, there are two ways to stress the usefulness of determining the domain and range. First, in many situations in physics, engineering and the natural sciences we derive equations for quantities like cost, weight or distance utilizing functions and algebraic expressions. Understanding the properties of an answer we attain for such quantities, like its domain and range, enables us to check the validity of our solution through physical intuition.

For instance, when using a function to calculate a quantity that must be strictly positive, like height or weight, if we are using a function whose range contains negative values, we should be wary. In some instances, this is a sign that we have improperly modeled the physical situation at hand. In others, it is a sign that the function we have computed is only valid on certain intervals, for example those values of \begin{align*}x\end{align*} which make the function positive. In order to remain consistent with the reality of the physical situation when our range extends to values that seem impossible, we often “restrict the domain”, meaning that we exclude the values of the input variable which lead to the impossible values of the range.

This Lesson contains a showcase of the graphs of many important types of functions that we will encounter throughout this textbook. A student should be able to identify these graphs quite easily in the first few weeks of class. They should be able to determine the intercepts and locations of important features of a function, such as the focus of a parabola, the center of a circle, and the domain of the logarithmic function. Most, if not all, of these topics should be review, but a strong understanding of these fundamentals will be important to developing the more complicated topics in this book.

In anticipating the next few lessons on limits and derivatives, it might be helpful to have students recognize that the function \begin{align*}y = |x|\end{align*} is unique amongst the functions showcased. All of the other functions except \begin{align*}|x|\end{align*} are smooth, meaning that they have no sharp corners or breaks in them. The absolute value of \begin{align*}x\end{align*} has the sharp corner at \begin{align*}x = 0\end{align*}, which is an example of a function having a point where the slope approaching from one side isn't equal to the slope when approaching from the other side. If one were to graph the slope of the function \begin{align*}f(x) = |x|\end{align*}, soon to be referred to as its derivative, we would find that the value \begin{align*}x = 0\end{align*} would be excluded, making the derivative of \begin{align*}|x|\end{align*} a discontinuous function.

The final topic brought up in this chapter is function transformation. This is an important technique for interpreting functions that arise in modeling physical situations to understand the behavior of systems without graphing them. Transformations allow us to take a prototypical function, like one of the \begin{align*}8\end{align*} showcased in the textbook, and alter their shape to get many different versions of them on the \begin{align*}x-y\end{align*} plane.

For instance, consider the parabola given by the formula \begin{align*}f(x) = x^2\end{align*}. What if we wanted to move the graph of our parabola to the right by \begin{align*}3 \;\mathrm{units}\end{align*}? As explained in this lesson, a rightward shift of \begin{align*}3\end{align*} would be enacted by subtracting \begin{align*}3\end{align*} from our variable, so instead of \begin{align*}f(x) = x^2\end{align*}, we would get \begin{align*}f(x) = (x-3)^2\end{align*}. The graphs of these functions are shown below.

An important example of where the shift transformation arises in a physical contexts the solution to the Wave Equation in two dimensions. In that case, if we were to start a wave on the middle of a string that had a particular shape \begin{align*}f(x)\end{align*}, we would get two copies of that wave, each half in amplitude, that move in opposite directions. This can be written as

\begin{align*}v(x) = \frac{1}{2} f (x+ct) + \frac{1}{2} f (x-ct)\end{align*}

This expression tells us that we have two copies of our original function \begin{align*}f(x)\end{align*}, divided in half in amplitude, with one copy shifted to the left by the product of c, the wave speed, and t, the time elapsed, and the other copy shifted similarly to the right. As time gets bigger, this shift grows larger, representing the wave moving away from its original position, and traveling along the string.

Transformations can create tremendous confusion for students because they appear in some ways the opposite of what one would expect. Take, for instance the shift of the function \begin{align*}f(x)\end{align*} to \begin{align*}f(x-c)\end{align*}. Many students will think that because we are subtracting \begin{align*}c\end{align*}, this corresponds to a shift in the negative direction. However, as we see above, by subtracting a constant \begin{align*}c\end{align*}, we actually shift the function in the positive direction.

Similarly, if we consider the transformation \begin{align*}f(x)\end{align*} to \begin{align*}f \left (\frac{x}{2} \right )\end{align*}, we might expect our original graph to be compressed by a factor of \begin{align*}2\end{align*}, since we are dividing by \begin{align*}2\end{align*}; conversely, if we consider the transformation of \begin{align*}f(x)\end{align*} to \begin{align*}f(2x)\end{align*}, we might expect our graph to be expanded by a factor of \begin{align*}2\end{align*}.

In each of these \begin{align*}3\end{align*} cases of function transformation, the opposite to what seems immediately apparent turns out to be true. If we transform our function \begin{align*}f(x)\end{align*} to \begin{align*}f(x-c)\end{align*}, we are shifting our function to the right by the value \begin{align*}c\end{align*}. Transforming \begin{align*}f(x)\end{align*} to \begin{align*}f \left (\frac{x}{2} \right )\end{align*} expands our original function by a factor of \begin{align*}2\end{align*}, and transforming \begin{align*}f(x)\end{align*} to \begin{align*}f(2x)\end{align*} compresses our original function by a factor of \begin{align*}2\end{align*}. These caveats should be emphasized at this stage to ensure that a student is able to easily identify how to graph common functions which have been transformed through these standard operations (called a shift, dilation and contraction, respectively). The rationale for these operations can be deduced algebraically.

Lesson 3: Modeling Data with Functions

In this lesson, students use their graphing calculators to find curves which best approximate a set of data points on a scatterplot. This technique is often referred to as “regression” or “curve-fitting”. Unlike traditional treatments of regression in statistics classes, which often focus exclusively on the topic of linear regression, Lesson 3 shows students that different sets of data are often best fit by a variety of different functions, depending on the visual character of the scatterplot. Though a linear approximation is sometimes the best approximation (and most often, the simplest), illustrating that we can also model data using higher order polynomials, trigonometric functions and transcendental functions may be new to many students.

The handling of real world data, even sets as small as provided in this Lesson, is usually handled by a computer or calculator since the calculations involved in determining the curve of best fit can be quite cumbersome. In this Lesson, calculating curves to fit the data is performed through both a graphing calculator as well as using Excel, and both are skills that a student should become comfortable with. It is important, however, to ensure that students understand the underlying reasoning their calculator is using to calculate curves of best fit since the criteria we can use to measure “best fit” can be interpreted very differently.

In the examples given, the lines of best fit are calculated by minimizing the least square error between the curve and the data points. This meaning that if we were to add up the distance squared between the curve selected by our curve fitting technique, and all of the data points, the curve that is selected will provide the smallest value for the sum of the squared error.

\begin{align*}\text{error} = \sum_{i=1}^M (f(x) - y_i)^2\end{align*}

This raises two important points that are hidden to the student if they exclusively rely on technology to find their curve. First, in some cases the use of the Least Squares approximation does not suit the purpose we are trying to achieve by fitting the data with a curve. Secondly, the Least Squares Approximation is so widely used because it is a Quadratic function, and thus is guaranteed to have a unique extreme value as illustrated in Lesson 2. This is helpful to only have a measurement of error that has only one unique maximum or minimum.

Take, for instance, the curves below which are both trying to approximate the same data set:

Clearly the red curve has the lower least square error, since the curve runs exactly through all of the points. Thus its total error is zero. But if we are trying to capture the trend of the data, however, the black curve is much better since it captures the undulations of the data, as well as the likely trend of the data at the end points. This is true even though it doesn't exactly fit most off the points. This is an example of where using the Least Square Error to fit our data may not be in our best interest, and is referred to as “over-fitting”.

In other cases, the presence of outlying and anomolous data may seriously affect the calculation of our regression line if we use the Least Squares Error as our criterion for curve fitting. If a student were to rely exclusively on the answers provided by their calculator, they may arrive at curves which are not appropriate to model the data they are interpreting.

An additional source of confusion for many students in understanding curve fitting is the difference between parameters and variables in our original function. For example, consider the standard equation of a line:

\begin{align*}y = mx + b\end{align*}

In this equation, the variables are \begin{align*}y\end{align*} and \begin{align*}x\end{align*}, and the parameters are the values \begin{align*}m\end{align*} and \begin{align*}b\end{align*}. For any particular line, \begin{align*}m\end{align*} and \begin{align*}b\end{align*} are fixed and determine the character of the line we are graphing. In the case of curve fitting, however, we now treat our values of \begin{align*}m\end{align*} and \begin{align*}b\end{align*} as variables, and try to find the values of these variables which optimize our problem in some sense (for instance, minimizing the Least Square Error). Once we find these values, we plug them into the values of \begin{align*}m\end{align*} and \begin{align*}b\end{align*} in the equation above, thus providing constant values for these parameters. For instance, on Page 29 the regression line found by the graphing calculator is \begin{align*}y = 0.76x + 14\end{align*}. To arrive at the values of \begin{align*}m = 0.76\end{align*} and \begin{align*}b = 14\end{align*} we had to allow these values to be variables and solve for them through a Least Squares technique. Once we have solved for them, they become constant values in our function.

Lesson 4: Introduction to The Calculus

Perhaps the most important topic in all of calculus is the concept of a limit, which is introduced in this Lesson through two of its most common uses: finding the slope of a tangent line to a curve at a given point (the derivative), and finding the area under a curve (the integral).

The tremendous usefuleness of a limit can be seen most clearly in the definition of the derivative, or finding the slope of a curve that is continuously changing, using what are referred to as secant line approximations. As discussed in this Lesson, the equation for the slope of a line is given by:

\begin{align*}\text{slope} = m = \frac{y_2 - y_1}{x_2 - x_1}\end{align*}

In determining the slope of the line that is tangent to the curve at a single point, we can interpret this as moving the two points between which we draw our secant line closer and closer to the point where we are looking for the tangent line to. To determine the slope at the point of interest, we can calculate the slopes of the secant line as we move its two endpoints closer and closer together.

A problem arises, however, when those two endpoints are brought together until their distance in the \begin{align*}x-\end{align*}direction, i.e. \begin{align*}x_2 - x_1\end{align*}, becomes zero, since it will lead us to divide by zero in the slope equation above. In most cases, this would make our slope fraction undefined, meaning it didn't have a slope. However just as the denominator becomes zero, the numerator also becomes zero, since now both values in the difference expression in the numerator, \begin{align*}f(x_2)\end{align*} and \begin{align*}f(x_1)\end{align*} become the same. We know that any fraction with a zero in it is equal to zero, but any fraction with a zero in the denominator is undefined; when both are true, we are left with an indeterminate situation. This is called “an indeterminate form”, and is what motivates most of the techniques developed in calculus.

It is problems like this which gave rise to the study of limits. Instead of considering the equation above when \begin{align*}x_2 = x_1\end{align*}, or when the secant endpoints' \begin{align*}x-\end{align*}values become exactly the same, we consider what happens in the limit as \begin{align*}x_2\end{align*} approaches \begin{align*}x_1\end{align*}. This is written as:

\begin{align*}\lim_{x_2 \to x_1} \frac{f(x_2) - f(x_1)}{x_2 - x_1}\end{align*}

Similarly, when we try to approximate the area under the curve as a set of rectangles, known as calculating Riemann sums, we must make our rectangles narrower and narrower to better approximate what the area under the curve actually looks like. This corresponds to making our rectangles “infinitisemally wide” (i.e. their width approaches zero), and so the expression for adding up the areas of these rectangles ALSO becomes indeterminate. The process of calculating this area in the presence of such indeterminancy is referred to as integration. Chapter 4 introduces the limiting operations needed to perform integration.

This lesson gives us a concrete example of how the technique of finding a derivative works by providing a table of values for a set of secant line approximations, and the sum of rectangular area approximations, as we move the endpoints of our interval closer and closer together. In these cases, the value that we are trying to determine (the slope of the secant lines, and the area of the sum of rectangles) start to converge towards a particular value. If this continues to happen as the distance between the endpoints gets smaller and smaller, then we say that the limit exists, and it is given by the value that these approximations approach.

It might be helpful for some students to show them examples where the above techniques don't work, particularly in cases where the derivative of a function does not exist, and thus the limit above does not exist. Take, for instance, the absolute value function \begin{align*}|x|\end{align*} we looked at in Lesson 2. If we were to construct a table of values for this situation, we would find that the slopes of our secant lines DON't approach the same value depending on how we move our two endpoints closer and closer together. This is a case where the limit of the above slope equation does NOT exist. These caveats will be discussed more thoroughly in Chapter 3.

Lesson 5: Finding Limits

The beginning of this Lesson takes students through the exercise of finding a limit using a table of values, but this time by using their graphing calculator on functions which would be much more difficult to evaluate by hand. In particular, we consider the function:

\begin{align*}f(x) = \frac{x+3}{x^2 + x - 6} = \frac{x+3}{(x-2)(x+3)}\end{align*}

An important point to raise here is the danger of relying on results from a graphing calculator or computer program, particularly when it comes to using limits. For the equation above, the calculator claims the value of the function is undefined at the two places where the denominator becomes zero, which may lead some students to believe that it doesn't have a limit as it approaches these points. But as we learned in the last lesson, this function actually can be evaluated in the limit as \begin{align*}x\end{align*} appoaches \begin{align*}-3\end{align*}, even though it would appear as if the zero in the denominator would be catastrophic.

The reason again that this function does have a limit as \begin{align*}x\end{align*} approaches \begin{align*}-3\end{align*} is that even though the denominator becomes zero, the numerator in this case also becomes zero. Therefore we have the indeterminate form \begin{align*}\frac{0}{0}\end{align*}, which may or may not have a finite value. It is important for students to understand that this is why we can have a limit at \begin{align*}x = -3\end{align*}, but we do not have a limit as we approach the other point where the denominator becomes zero, \begin{align*}x = 2\end{align*}. When \begin{align*}x = 2\end{align*}, the expression becomes \begin{align*}\frac{5}{0}\end{align*} which is clearly undefined.

As mentioned in the previous Lesson, it can be helpful for many students to be shown examples of when limits do NOT exist in order to understand how and when they are useful. In the case above, we considered the case where the derivative of a function does not exist at a particular point as motivated by the formula for the derivative which incorporates the limit. In that case, \begin{align*}|x|\end{align*}, the function DID exist at the point \begin{align*}x = 0\end{align*}, but the derivative didn't. In the function above, at the point \begin{align*}x = 2\end{align*} the function itself does not exist, an example of a function being discontinuous. This can be seen very clearly from the graph of the function, since as we approach the point of interest, \begin{align*}x = 2\end{align*}, from both sides, the value of our function approaches very different values. As we move towards \begin{align*}x = 2\end{align*} from the negative side, our function approaches negative infinity, whereas from the positive side, it approaches positive infinity.

This example offers a good opportunity to present the formal definition of a limit provided at the end of this Lesson. This definition presents difficulty for even advanced students of mathematics, and is primarily used for the topic of real analysis. Intuitively, we can understand the formal definition of the limit in the context of our function which does not have a limit near the point \begin{align*}x = 2\end{align*}. The definition of a limit tells us that if we are in an arbitrarily small interval around \begin{align*}x = 2\end{align*}, then we should be able to find an arbitrarily small interval in the \begin{align*}y-\end{align*}direction in which our function must be. Clearly if we were to look at an interval around \begin{align*}x =2\end{align*}, however, our function is not contained in an arbitrarily small interval, since it approaches negative infinity on one side of \begin{align*}x = 2\end{align*}, and positive infinity on the other side of \begin{align*}x = 2\end{align*}. There would not be an interval in the \begin{align*}y-\end{align*}direction around the function at \begin{align*}x = 2\end{align*} which would be able to bound the entire function. Hence the limit does not exist.

Lesson 6: Evaluating Limits

As mentioned in the last Lesson, the use of the formal definition of a limit can be quite cumbersome, and is rarely used in most situations that a student will encounter. It will be helpful in understanding theorems and concepts later in Calculus and Analysis, but the actual task of finding a limit is usually much more straightforward. Sometimes it can be as simple as performing direct substitution into our algebraic expression as we saw in Lesson 4. But in most real applications, more complicated techniques are required.

In this Lesson, the student is introduced to some important properties of limits which will enable them to use the technique of substitution with more complicated algebraic expressions. Although the properties of limits being additive should seem intuitive, it should be noted that the truth of these properties in the presence of multiplication, division and raising expressions to exponents is much more subtle. In each of these cases, it can be seen that the operation of taking a limit is commutative, distributive and associative over multiplication and addition. This enables us to use substitution to find the limit for arbitrary polynomial expressions by substituting our value of interest for the variable in each term.

The next two techniques to finding limits will appear much more strange to students, especially since the motivation of how and why they work is reserved for later chapters when the concept of a derivative has been introduced more thoroughly. The important point to stress here is that the simple technique of substitution is often insufficient to finding limits of many important functions, and more sophisticated techniques exist in many of these cases.

The first such technique entails finding limits to rational expressions when they take on an indeterminate form. Indeterminate forms are examples of where substitution is not sufficient to finding limits of expressions, since when we have a zero divided by another zero, or infinity divided by another infinity we cannot be sure what the value of our substituted algebraic expression is. This lesson exposes students to handling such situations more formally, and introduces them to the concept of a “removable singularity”. Even though the denominator does become zero at the point of interest, because the numerator also becomes zero in exactly the same way, we can remove the denominator by factoring the numerator and cancelling.

The second technique introduced in this Lesson as a way to calculate more complicated limits is called the Squeeze Theorem. The Squeeze Theorem states that if there exist two functions which bound our function of interest, meaning that our function is always in between the two other functions, and those functions both converge to the same limit at a point, then our function must also converge to that limit. What the Squeeze Theorem is saying is that we are squeezing our function in between two functions that are approaching the same value. Since they always bound the function of interest, the function of interest must necessarily also be that value, since there are no other values for it to be in between the functions that are squeezing it.

Lesson 7: Continuity

This Lesson builds on the previous discussion of limits to introduce the notion of a function being continuous or discontinuous. What we find is that even if a function has a limit at a particular point, that does not mean that it is defined at that point. This idea is illustrated by returning to the example of a rational function given above:

\begin{align*}f(x) = \frac{x+3}{x^2 + x -6} = \frac{x+3}{(x-2)(x+3)}\end{align*}

As students learned in Lesson 5, the above expression does have a limit as \begin{align*}x\end{align*} approaches \begin{align*}-3\end{align*}, even though the denominator is equal to zero at this point. This is seen clearly in the factored form of the expression, where the terms \begin{align*}x +3\end{align*} will cancel each other leaving no term in the denominator equal to zero. This type of discontinuity is referred to as a removable singularity. On the other hand, this function does NOT approach a limit as \begin{align*}x\end{align*} approaches \begin{align*}+2\end{align*}, for reasons discussed above.

Near the point \begin{align*}x = -3\end{align*}, the function behaves rather smoothly, because it does have a limit as it approaches \begin{align*}-3\end{align*} even though at that point it is undefined. This is represented graphically by a curve with a hole punched out at the point of discontinuity - this is called a point discontinuity. This example illustrates that just because a limit exists as we approach a certain point, it does not mean that the function itself has to exist at that point.

At the point \begin{align*}x = 2\end{align*}, however, our discontinuity appears very different. In this case, we have a discontinuity that is manifested as our curve approaching positive infinity and negative infinity in opposite directions depending on the side of \begin{align*}x = 2\end{align*} that we are on. At \begin{align*}x = 2\end{align*}, there exists a “vertical asymptote” at the point of discontinuity. This is also a very common occurrence for rational expressions which appear in physical situations. In this case \begin{align*}x = 2\end{align*} is referred to as a pole.

The important point to note is that we can have different types of discontinuities, and we often use limits to determine which type we have. This helps us to understand the behavior of our function around points of interest, like the places where the denominator equals zero, without the trouble of graphing the function.

The final topic discussed in Lesson 7 is one-sided limits, which you may have already addressed in the context of the absolute value function, \begin{align*}f(x) = |x|\end{align*}, in Lesson 2. In that discussion it was noted that as we considered the slope of our function as we approached \begin{align*}x = 0\end{align*} from the left, our slope was \begin{align*}-1\end{align*}. If we approached \begin{align*}x = 0\end{align*} from the other side, our slope was \begin{align*}+1\end{align*}. Because our slopes didn't match, and abruptly changed from \begin{align*}-1\end{align*} to \begin{align*}+1\end{align*}, the slope of our line is undefined at \begin{align*}x = 0\end{align*}. If, however, we defined the existence of a limit as only needing to be valid as we approached from one side, we would say that the slope DOES have two one-sided limits, just not a two-sided limit.

In cases like the square root function shown in the text, it is seen that because our function is only defined on one side of the point \begin{align*}x = 0\end{align*}, we would have to conclude that the function square root of \begin{align*}x\end{align*} does not have a limit there, since we cannot approach from the left. In such cases, the one-sided right hand limit (i.e. approaching from the right) does exist, but the one-sided left handed limit (i.e. approaching from the left) does not.

Lesson 8: Infinite Limits

The last type of limits treated in Chapter 1 are those where the value we are approaching is either infinity or negative infinity. This allows us to determine the behavior of our function as it extends infinitely in both directions along the \begin{align*}x-\end{align*}axis. By understanding the behavior of our function as it approaches positive and negative infinity, we will be able to without having to plug in multiple values as \begin{align*}x\end{align*} gets very large in the positive and negative directions.

Up until this point, the only example of an indeterminate form that students have been exposed to is \begin{align*}\frac{0}{0}\end{align*}. We noted in those cases that we could not determine an answer for our expression by mere inspection, since most fractions with a zero in the numerator are equal to zero, whereas a most fractions with a zero in the denominator are undefined.

Which one of these facts wins out in such cases? This is where the techniques of finding limits proved useful.

Similarly, what if we are presented with a fraction where the numerator is approaching infinity, and the denominator is approaching infinity?? This is another situation in which we are presented with an indeterminate form. A fraction whose numerator gets bigger and bigger will grow larger and larger, but if its denominator is also growing larger and larger, then the value of the fraction will get smaller and smaller. These are important points to stress to students, since it may be one of the first times that they have considers what happens to an expression when the variable gets infinitely large. It is an essential tool to analyzing many important physical situations.

An important point to stress to students, and to make sure that they understand, is that as the denominator of a fraction approaches positive or negative infinity, the value of the overall fraction approaches zero. This is a point that can be stressed by looking at some examples of fractions with large numbers in the denominator. Take for instance, the sequence:

\begin{align*}\frac{1}{10} & = 0.1\\ \frac{1}{100} & = 0.01\\ \frac{1}{1000} & = 0.001\\ & \ldots\end{align*}

As the denominator gets larger, the value of our fraction gets smaller. We say that in the limit as our denominator becomes infinite, the value of our fraction becomes “infinitesimal”. Being able to quickly understand rational expressions that have complicated algebraic terms in them, and to identify what happens when the independent variable gets very large, is an important tool in understanding physical situations.

An example of this is the determination of steady state behavior of a physical system. In many cases, the behavior of a physical system can be modeled as an exponential function, such as

\begin{align*}f(x) = e^{ax}\end{align*}

where a is a constant. Let's consider the case where a is positive, and our independent variable \begin{align*}``x"\end{align*} is a measure of time. For instance, suppose we pluck a string on a musical instrument, and listen to the sound over time. What happens to the sound as time goes on for longer and longer?

In the case where a is positive, if we tak\begin{align*}3e\end{align*} the limit of the expression above as \begin{align*}x\end{align*} goes to infinity, we find that our function also grows to infinity! This is true, since a positive value greater than \begin{align*}1\end{align*}, when raised to a large positive power, grows larger and larger as we raise it to larger and larger powers.

However, what happens if a is negative? Now, as \begin{align*}x\end{align*} goes to infinity, the expression goes to zero, since:

\begin{align*}\lim_{t \to \infty} e^{-at} = \lim_{t \to \infty} \frac{1}{e^{at}} \rightarrow \frac{1}{\infty} \rightarrow 0\end{align*}

These behaviors are observed in the graphs of these functions.

Often when we are designing mechanical or electrical systems, the type and quantity of materials we use will affect the value of the constant in the exponential term that describes our system. By designing the system in such a way that this constant is negative, we guarantee that after a long time, any input to our system eventually dies away. If we designed the system in such a way that the exponent was POSITIVE, we would find that any input to our system would make its output start to grow very very large, without any bound. This is referred to as an “unstable” system, and the task of design then becomes to make our exponents negative, the importance of which is seen by taking the limit of our expression as time goes to infinity.

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