Suppose you stand exactly 4 feet from a wall, and begin moving toward the wall by halving the distance remaining with each step. How many steps would it take to actually get to the wall? How far would you walk in the process?
This is a question of limit, see if you can puzzle it out before the review at the end of this lesson.
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Khan Academy: Introduction to Limits (HD)
Guidance
Limits and Asymptotes
Consider the function \begin{align*}f(x)=\frac{1}{x}\end{align*}. A graph of this function is shown below.
Notice that as the values of x get larger and larger, the graph gets closer and closer to the x-axis. In terms of the function values, we can say that as x gets larger and larger, f(x) gets closer and closer to 0. Formally, this kind of behavior of a function is called a limit. We say that as x approaches infinity, the limit of the function is 0. The line y = 0 is called the asymptote of the graph, it represents the value that f(x) will never quite reach. We can also say that @$\begin{align*} f(x)=\frac{1}{x}\end{align*}@$ is asymptotic to the line y = 0.
If we consider the behavior of the function as x approaches @$\begin{align*}-\infty\end{align*}@$, we see the same result: the limit of the function has x approaches @$\begin{align*}-\infty\end{align*}@$ is also 0. Notice that this has the same asymptote: y = 0.
To be even more formal, we can write limits using a special notation. For the first limit, we write: @$\begin{align*} \lim_{x \to \infty} f(x) = 0.\end{align*}@$ For the second limit, we write @$\begin{align*} \lim_{x \to -\infty} f(x) = 0.\end{align*}@$ For any limit, here we will always write the x under the abbreviation “lim”, and then we will write the function under consideration. We can also write each of these limits with the specific function:
@$\begin{align*} \lim_{x \to \infty} \frac{1} {x} = 0\end{align*}@$ and @$\begin{align*} \lim_{x \to -\infty} \frac{1} {x} = 0\end{align*}@$.
Because we are focused on end behavior, we are considering the limit of functions as x approaches @$\begin{align*}\pm \infty\end{align*}@$, and so the asymptotes we will find are horizontal lines. If we were examining other aspects of functions, we might find asymptotes that are vertical lines. For example, the function @$\begin{align*} f(x)=\frac{1}{x}\end{align*}@$ has a vertical asymptote at x = 0, or the y-axis. That is, the graph approaches the y-axis, as x values get closer and closer to 0.
Example A
Write the limit described using limit notation.
- The limit of some function @$\begin{align*}f(x)\end{align*}@$ as x approaches infinity is 2.
Solution: We write the limit as follows:
- @$\begin{align*}\lim_{x \to \infty} f(x) = 2\end{align*}@$
Example B
Explain in words the meaning of the limit statement: @$\begin{align*}\lim_{x \to \infty} \left (3 + \frac{2} {x}\right) = 3\end{align*}@$
Solution:
@$\begin{align*}\lim_{x \to \infty} \left (3 + \frac{2} {x}\right) = 3\end{align*}@$ means: "As larger and larger numbers are substituted in for @$\begin{align*}x\end{align*}@$ in the function @$\begin{align*}3+2/x\end{align*}@$, the value comes closer and closer to @$\begin{align*}3\end{align*}@$.
This is due to the fact that the value added to @$\begin{align*}3\end{align*}@$ gets smaller and smaller, down to effectively @$\begin{align*}0\end{align*}@$ as @$\begin{align*}2\end{align*}@$ is divided by larger and larger numbers.
Example C
Determine the horizontal asymptote of the function @$\begin{align*}g(x)=\frac{2x-1}{x}\end{align*}@$ and express the asymptotic relationship using limit notation.
Solution:
This function is asymptotic to the line y = 2. The limit is written as @$\begin{align*}\lim_{x \to \neq \infty} \frac{2x - 1} {x} = 2\end{align*}@$.
We can determine the asymptote (and hence the limit) if we look at the graph. However, we can also analyze the equation to determine the limit. Consider the function @$\begin{align*}g(x)=\frac{2x-1}{x}\end{align*}@$. As x approaches infinity, the x values are getting larger and larger. For sufficiently large values of x, the values of the expression 2x - 1 are very close to the values of the expression 2x, because subtracting one from a large number is fairly insignificant. Thus for sufficiently large values of x, @$\begin{align*} \frac{2x-1}{x} \approx \frac{2x}{x} \approx 2 \end{align*}@$. As you can see from the accompanying table, which was created by a TI-83 graphing calculator, the function value gets closer to 2 as we look at larger and larger x values.
Concept question wrap-up: You will recall the question at the beginning of the lesson about the distance involved in a strange walk towards a wall. "If you start 4ft from a wall, and halve the distance to it with each step, how many steps will it take, and how far will you walk, before you actually touch the wall?" Logically, we know that there is only a total distance of 4 feet between you and the wall, so no matter how you break it up, you cannot walk more than 4 feet. However, the actual distance you cover, and the number of steps it would take, cannot truly be defined since there could always be 1/2 of the remaining distance left. Technically speaking, you could continue the process forever without actually touching the wall! Of course, in practice, your ability to only move 1/2 of the remaining distance is limited by muscle control and measurement accuracy, so you would touch the wall before very many steps were actually taken. Mathematically: @$\begin{align*}\lim_{n \to \infty} \left (4 - \frac{4} {2^n}\right) = 4\end{align*}@$ Where @$\begin{align*}n\end{align*}@$ is the number of steps. |
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In other words: "As the remaining distance gets closer and closer to 0, the total distance approaches 4" |
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Guided Practice
Describe each case given below and sketch a graph of each function with the given properties:
- 1) @$\begin{align*}\lim_{x \to \infty} f(x) = 2\end{align*}@$
- 2) @$\begin{align*}\lim_{x \to -\infty} f(x) = 0\end{align*}@$
- 3) @$\begin{align*}\lim_{x \to 0+} f(x) = \infty\end{align*}@$
- 4) @$\begin{align*}\lim_{x \to 4} f(x) = -\infty\end{align*}@$
- 5) @$\begin{align*}\lim_{x \to 4} f(x) = 3\end{align*}@$
Answers
There are a number of possible graphs for each of the required cases above, one example of each is offered below.
- 1) This reads: "The limit of f(x) as x approaches infinity is 2." In other words, as x gets very big, f(x) or y gets infinitely close to 2.
- 2) This reads: "The limit of f(x) as x approaches negative infinity is 0." In other words, as x gets massively negative, f(x) or y gets infinitely close to 0.
- 3) This reads: "The limit of f(x) as x approaches zero from the positive direction is infinite." In other words, as x approaches from the right side of the graph, and gets infinitely close to zero, f(x) or y grows infinitely large.
- 4) This reads: "The limit of f(x) as x approaches 4 is negative infinity." In other words, as x gets infinitely close to 4, f(x) or y gets infinitely negative.
- 5) This reads: "The limit of f(x) as x approaches 4 is 3." In other words, as x gets infinitely close to 4, f(x) or y gets infinitely close to 3. This can be a straight line, as y approaches 3 when 4 approaches 4 from either direction.
Explore More
- Define the terms horizontal asymptote and vertical asymptote.
- Explain the difference between @$\begin{align*}\lim_{x \to -6} f(x) = \infty\end{align*}@$ and @$\begin{align*}\lim_{x \to \infty} f(x) = -6\end{align*}@$.
- Explain what @$\begin{align*}\lim_{x \to \infty} f(x) = 200\end{align*}@$ means.
- Explain what @$\begin{align*}\lim_{x \to 175} f(x) = 175\end{align*}@$ means.
Evaluate the following limits, if they exist. If a limit does not exist, explain why.
- @$\begin{align*}\lim_{t \to \infty}\frac{3t^2 - 7t} {t-8}\end{align*}@$
- @$\begin{align*}\lim_{t \to \infty} 3\end{align*}@$
- @$\begin{align*}\lim_{t \to \infty}(t^2 - t^4)\end{align*}@$
- @$\begin{align*}\lim_{x \to \infty} x +\sqrt {x^2 + 2x}\end{align*}@$
- Find the horizontal and vertical asymptotes of the following function: @$\begin{align*}h(g) = \frac {5g^2 - 7g +9}{g^2 - 2g -3}\end{align*}@$
Given: @$\begin{align*} f(x) = \frac{x^2 - x - 6}{x^2 - 2x - 8}\end{align*}@$ perform the following:
- Find the horizontal and vertical asymptotes. Determine the behavior of @$\begin{align*} f \end{align*}@$ near the vertical asymptotes.
- Find the roots, y intercept and “holes” in the graph.
Determine @$\begin{align*}\lim_{t \to \infty} \frac{1}{t^n}\end{align*}@$ if:
- @$\begin{align*}n > 0\end{align*}@$
- @$\begin{align*}n < 0\end{align*}@$
- @$\begin{align*}n = 0\end{align*}@$
Let G & H be polynomials. Find @$\begin{align*}\lim_{x \to \infty} \frac{G(x)}{H(x)}\end{align*}@$ if:
- The degree of G is less than the degree of H
- The degree of G is greater than the degree of H
- The degree of G is the same as the degree of H
- A pool contains 8000 L of water. An additive that contains 30g of salt per liter of water is added to the pool at a rate of 25 L per minute. a) Show that the concentration of salt after t minutes in grams per liter is: @$\begin{align*}C(t) = \frac{(t)30g\cdot25}{8000l + 25(t)l}\end{align*}@$ b) What happens to the concentration as time increases to @$\begin{align*}\infty\end{align*}@$? Physically, why does this make sense?