1.5: Finding Limits
Learning Objectives
A student will be able to:
 Find the limit of a function numerically.
 Find the limit of a function using a graph.
 Identify cases when limits do not exist.
 Use the formal definition of a limit to solve limit problems.
Introduction
In this lesson we will continue our discussion of the limiting process we introduced in Lesson 1.4. We will examine numerical and graphical techniques to find limits where they exist and also to examine examples where limits do not exist. We will conclude the lesson with a more precise definition of limits.
Let’s start with the notation that we will use to denote limits. We indicate the limit of a function as the
So, in the example from Lesson 1.3 concerning the function
Recall also that we found that the slope values tended to the value
Finding Limits Numerically
In our example in Lesson 1.3 we used this approach to find that
Consider the rational function
Unlike our simple quadratic function,
Even though the function is not defined at
Can you guess the value of
Press 2ND [TBLSET] and change Indpnt from Auto to Ask. Now when you go to the table, enter
Finding Limits Graphically
Let’s continue with the same problem but now let’s focus on using the graph of the function to determine its limit.
We enter the function in the
Our graph above is set to the normal viewing window
The graphing calculator will allow us to calculate limits graphically, provided that we have the function rule for the function so that we can enter its equation into the calculator. What if we have only a graph given to us and we are asked to find certain limits?
It turns out that we will need to have pretty accurate graphs that include sufficient detail about the location of data points. Consider the following example.
Example 1:
Find
By inspection, we see that as we approach the value
Nonexistent Limits
We sometimes have functions where
What do you think the limit will be as we let
Our inspection of the graph suggests that the function around
For this example, we say that
Formal Definition of a Limit
We conclude this lesson with a formal definition of a limit.
Definition:

We say that the limit of a function
f(x) ata isL, written aslimx→af(x)=L , if for every open intervalD ofL, there exists an open intervalN ofa, that does not includea, such thatf(x) is inD for everyx inN. 
This definition is somewhat intuitive to us given the examples we have covered. Geometrically, the definition means that for any lines
y=b1,y=b2 below and above the liney=L , there exist vertical linesx=a1,x=a2 to the left and right ofx=a so that the graph off(x) betweenx=a1 andx=a2 lies between the linesy=b1 andy=b2 . The key phrase in the above statement is “for every open intervalD ”, which means that even ifD is very, very small (that is,f(x) is very, very close toL ), it still is possible to find intervalN wheref(x) is defined for all values except possiblyx=a .
Example 2:
Use the definition of a limit to prove that
We need to show that for each open interval of
Equivalently, we must show that for every interval of
The first inequality is equivalent to
Hence if we take
Fortunately, we do not have to do this to evaluate limits. In Lesson 1.6 we will learn several rules that will make the task manageable.
Lesson Summary
 We learned to find the limit of a function numerically.
 We learned to find the limit of a function using a graph.
 We identified cases when limits do not exist.
 We used the formal definition of a limit to solve limit problems.
Multimedia Links
For another look at the definition of a limit, the series of videos at Tutorials for the Calculus Phobe has a nice intuitive introduction to this fundamental concept (despite the whimsical name). If you want to experiment with limits yourself, follow the sequence of activities using a graphing applet at Informal Limits. Directions for using the graphing applets at this very useful site are also available at Applet Intro.
Review Questions
 Use a table of values to find
limx→−2x2−4x+2 . Use
x− values ofx=−1.9,−1.99,−1.999,−2.1,−2.099,−2.0099. don't purge me  What value does the sequence of values approach?
 Use
 Use a table of values to find
limx→122x−12x2+3x−2 . Use
x− values ofx=.49,.495,.49999,.51,.5099,.500001.  What value does the sequence of values approach?
 Use
 Consider the function
p(x)=3x3−3x. Generate the graph ofp(x) using your calculator. Find each of the following limits if they exist. Use tables with appropriatex values to determine the limits.
limx→4(3x3−3x) don't purge me 
limx→−4(3x3−3x) 
limx→0(3x3−3x) don't purge me  Find the values of the function corresponding to
x=4,−4,0. How do these function values compare to the limits you found in #ac? Explain your answer.

 Examine the graph of
f(x) below to approximate each of the following limits if they exist.
limx→3f(x) 
limx→2f(x) 
limx→1f(x) 
limx→4f(x)

 Examine the graph of
f(x) below to approximate each of the following limits if they exist.
limx→2f(x) 
limx→0f(x) 
\begin{align*}\lim_{x \to 4} f(x)\end{align*}
limx→4f(x) 
\begin{align*}\lim_{x \to 50} f(x)\end{align*}
limx→50f(x)

In problems #68, determine if the indicated limit exists. Provide a numerical argument to justify your answer.

\begin{align*}\lim_{x \to 2} (x^2 + 3) \end{align*}
limx→2(x2+3) don't purge me 
\begin{align*}\lim_{x \to 1} \frac{x + 1} {x^2  1}\end{align*}
limx→−1x+1x2−1 
\begin{align*}\lim_{x \to 2} \sqrt{ 2x + 5} \end{align*}
limx→2−2x+5−−−−−−−√ don't purge me
In problems #910, determine if the indicated limit exists. Provide a graphical argument to justify your answer. (Hint: Make use of the [ZOOM] and [TABLE] functions of your calculator to view functions values close to the indicated \begin{align*}x\end{align*}

\begin{align*}\lim_{x \to 4} (x^2 + 3x) \end{align*}
limx→4(x2+3x) don't purge me 
\begin{align*}\lim_{x \to 1} \frac{x + 1} {x + 1}\end{align*}
limx→−1x+1x+1 don't purge me
Review Answers

 The corresponding \begin{align*}y\end{align*}
y values are 3.9, 3.99, 3.999, 4.1, 4.099, and 4.0099. 
\begin{align*}\lim_{x \to 2} \frac{x^2  4} {x + 2} = 4\end{align*}
limx→−2x2−4x+2=−4
 The corresponding \begin{align*}y\end{align*}
y values are 0.401606426, 0.400801603, 0.4000016, 0.398406375, 0.398422248, and 0.39999984. 
\begin{align*}\lim_{x \to \frac{1} {2}} \frac{2x  1} {2x^2 + 3x  2} = \frac{2} {5}\end{align*}
limx→122x−12x2+3x−2=25

\begin{align*}\lim_{x \to 4} (3x^3  3x) = 180\end{align*}
limx→4(3x3−3x)=180  \begin{align*}\lim_{x \to 4} (3x^3  3x) = 180\end{align*}
 \begin{align*}\lim_{x \to 0} (3x^3  3x) = 0\end{align*}
 They are the same values because the function is defined for each of these \begin{align*}x\end{align*}values.
 \begin{align*}\lim_{x \to 3} f(x) = 1.5\end{align*}
 \begin{align*}\lim_{x \to 2} f(x) = 0\end{align*}
 \begin{align*}\lim_{x \to 1} f(x) = 2\end{align*}
 \begin{align*}\lim_{x \to 4} f(x)\end{align*} does not exist.
 \begin{align*}\lim_{x \to 2} f(x) = 0\end{align*}
 \begin{align*}\lim_{x \to 0} f(x)\end{align*} does not exist.
 \begin{align*}\lim_{x \to 4} f(x)\end{align*} is some number close to \begin{align*}1\end{align*} and less than \begin{align*}1\end{align*}, but not equal to \begin{align*}1\end{align*}.
 \begin{align*}\lim_{x \to 50} f(x)\end{align*} is some number close to \begin{align*}1\end{align*} and less than \begin{align*}1\end{align*}, but not equal to \begin{align*}1\end{align*}.
 The corresponding \begin{align*}y\end{align*}
 The limit does exist. This can be verified by using the [TRACE] or [TABLE] function of your calculator, applied to \begin{align*}x\end{align*} values very close to \begin{align*}x = 2\end{align*}.
 The limit does exist. This can be verified by using the [TRACE] or [TABLE] function of your calculator, applied to \begin{align*}x\end{align*} values very close to \begin{align*}x = 1\end{align*}.
 The limit does exist. This can be verified by using the [TRACE] or [TABLE] function of your calculator, applied to \begin{align*}x\end{align*} values very close to \begin{align*}x = 2\end{align*}.
 The limit does exist. This can be verified with either the [TRACE] or [TABLE] function of your calculator.
 The limit does not exist; [ZOOM] in on the graph around \begin{align*}x = 1\end{align*} and see that the \begin{align*}y\end{align*}values approach a different value when approached from the right and from the left.
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