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Evaluate Limits Graphically and Vertical Asymptotes

Discontinuities' and asymptotes' effect on limits.

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Practice Evaluate Limits Graphically and Vertical Asymptotes
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Graphs to Find Limits

A limit can describe the end behavior of a function. This is called a limit at infinity or negative infinity. A limit can also describe the limit at any normal  $x$ value. Sometimes this is simply the height of the function at that point. Other times this is what you would expect the height of the function to be at that point even if the height does not exist or is at some other point. In the following graph, what are $f(3), \ \lim_{x \to 3}f(x), \ \lim_{x \to \infty}f(x)$ ?

Watch This

http://www.youtube.com/watch?v=LdewtuWi7fM James Sousa: Determining Basic Limits Graphically

Guidance

When evaluating the limit of a function from its graph, you need to distinguish between the function evaluated at the point and the limit around the point.

Functions like the one above with discontinuities, asymptotes and holes require you to have a very solid understanding of how to evaluate and interpret limits.

At $x=a$ , the function is undefined because there is a vertical asymptote. You would write:

$f(a)=DNE, \ \lim_{x \to a}f(x)=DNE$

At $x=b$ , the function is defined because the filled in circle represents that it is the height of the function. This appears to be at about 1. However, since the two sides do not agree, the limit does not exist here either.

$f(b)=1, \ \lim_{x \to b}f(x)=DNE$

At $x=0$ , the function has a discontinuity in the form of a hole. It is as if the point  $(0,-2.4)$ has been lifted up and placed at $(0,1)$ . You can evaluate both the function and the limit at this point, however these quantities will not match. When you evaluate the function you have to give the actual height of the function, which is 1 in this case. When you evaluate the limit, you have to give what the height of the function is supposed to be based solely on the neighborhood around 0. Since the function appears to reach a height of -2.4 from both the left and the right, the limit does exist.

$f(0)=1,\ \lim_{x \to 0}f(x)=-2.4$

At $x=c$ , the limit does not exist because the left and right hand neighborhoods do not agree on a height. On the other hand, the filled in circle represents that the function is defined at  $x=c$ to be -3.

$f(c)=-3, \ \lim \limits_{x \to c}f(x)=DNE$

At  $x \rightarrow \infty$ you may only discuss the limit of the function since it is not appropriate to evaluate a function at infinity (you cannot find $f(\infty)$ ). Since the function appears to increase without bound, the limit does not exist.

$\lim \limits_{x \to \infty}f(x)=DNE$

At $x \rightarrow - \infty$ the graph appears to flatten as it moves to the left. There is a horizontal asymptote at  $y=0$ that this function approaches as $x \rightarrow - \infty$ .

$\lim \limits_{x \to - \infty}f(x)=0$

When evaluating limits graphically, your main goal is to determine whether the limit exists. The limit only exists when the left and right sides of the functions meet at a specific height. Whatever the function is doing at that point does not matter for the sake of limits. The function could be defined at that point, could be undefined at that point, or the point could be defined at some other height. Regardless of what is happening at that point, when you evaluate limits graphically, you only look at the neighborhood to the left and right of the function at the point.

Example A

Evaluate the following expressions using the graph of the function $f(x)$ .

1. $\lim_{x \to - \infty}f(x)$
2. $\lim_{x \to -1}f(x)$
3. $\lim_{x \to 0}f(x)$
4. $\lim_{x \to 1}f(x)$
5. $\lim_{x \to 3}f(x)$
6. $f(-1)$
7. $f(2)$
8. $f(1)$
9. $f(3)$

Solution:

1. $\lim_{x \to - \infty}f(x)=0$
2. $\lim_{x \to -1}f(x)=DNE$
3. $\lim_{x \to 0}f(x)=-2$
4. $\lim_{x \to 1}f(x)=0$
5. $\lim_{x \to 3}f(x)=DNE$ (This is because only one side exists and a regular limit requires both left and right sides to agree)
6. $f(-1)=0$
7. $f(0)=-2$
8. $f(1)=2$
9. $f(3)=0$

Example B

Sketch a graph that has a limit at $x=2$ , but that limit does not match the height of the function.

Solution: While there are an infinite number of graphs that fit this criteria, you should make sure your graph has a removable discontinuity at $x=2$ .

Example C

Sketch a graph that is defined at  $x=-1$ but  $\lim_{x \to -1}f(x)$ does not exist.

Solution: The graph must have either a jump or an infinite discontinuity at  $x=-1$ and also have a solid hole filled in somewhere on that vertical line.

Concept Problem Revisited

Given the graph of the function  $f(x)$ to be:

$\lim \limits_{x \to 3}f(x) =\frac{1}{2}$

$f(3)=3$

$\lim_{x \to \infty}f(x) =DNE$

Vocabulary

The phrase “does not exist” or DNE is used with limits to imply that the limit does not approach a particular numerical value. Sometimes this means that the limit continues to grow bigger or smaller to infinity. Sometimes it means that the limit can’t decide between two disagreeing values. There are times when instead of indicating the limit does not exist, you might choose to write the limit goes to infinity. This notation is technically not correct, but it does provide more useful information than writing DNE and therefore can be acceptable.

Guided Practice

1. Identify everywhere where the limit does not exist and where the limit does exist in the following function.

2. Evaluate and explain how to find the limits as  $x$ approaches 0 and 1 in the previous question.

3. Evaluate the limits of the following piecewise function at -2, 0 and 1.

$f(x) = \begin{cases} 2 & x<-2 \\-1 & x=-2 \\-x-2 & -2 < x \le 0\\x^2 & 0 < x < 1 \\-2 & x=1 \\x^2 & 1 < x\end{cases}$

1. The limit does not exist at $x=-3,-1,2,+ \infty,- \infty$ . At every other point (including $x=1$ ), the limit does exist.

2. $\lim_{x \to 0}f(x)=2, \ \lim_{x \to 1}f(x)=1$

Both of these limits exist because the left hand and right hand neighborhoods of these points seem to approach the same height. In the case of the point $(0,2)$  the function happened to be defined there. In the case of the point $(1,1)$  the function happened to be defined elsewhere, but that does not matter. You only need to consider what the function does right around the point.

3. Since you already know how to graph piecewise functions (graph each function in the  $x$ interval indicated) you can then observe graphically the limits at -2, 0 and 1.

$\lim \limits_{x \to -2}f(x)=2$

$\lim \limits_{x \to 0}f(x)=DNE$

$\lim \limits_{x \to 1}f(x)=1$

Practice

Use the graph of $f(x)$  below to evaluate the expressions in 1-6.

1. $\lim_{x \to - \infty}f(x)$

2. $\lim_{x \to \infty}f(x)$

3. $\lim_{x \to 2}f(x)$

4. $\lim_{x \to 0}f(x)$

5. $f(0)$

6. $f(2)$

Use the graph of  $g(x)$ below to evaluate the expressions in 7-13.

7. $\lim_{x \to - \infty}g(x)$

8. $\lim_{x \to \infty}g(x)$

9. $\lim_{x \to 2}g(x)$

10. $\lim_{x \to 0}g(x)$

11. $\lim_{x \to 4}g(x)$

12. $g(0)$

13. $g(2)$

14. Sketch a function  $h(x)$ such that $h(2)=4$ , but $\lim_{x \to 2}h(x)=DNE$ .

15. Sketch a function  $j(x)$ such that $j(2)=4$ , but $\lim_{x \to 2}j(x)=3$ .

Vocabulary Language: English

Asymptotes

Asymptotes

An asymptote is a line on the graph of a function representing a value toward which the function may approach, but does not reach (with certain exceptions).
discontinuities

discontinuities

The points of discontinuity for a function are the input values of the function where the function is discontinuous.
DNE

DNE

The phrase “does not exist” or “DNE” is used with limits to imply that the limit does not approach a particular numerical value. Sometimes this means that the limit continues to grow bigger or smaller to infinity. Sometimes it means that the limit can’t decide between two disagreeing values.
does not exist

does not exist

The phrase “does not exist” or “DNE” is used with limits to imply that the limit does not approach a particular numerical value.
End behavior

End behavior

End behavior is a description of the trend of a function as input values become very large or very small, represented as the 'ends' of a graphed function.
Hole

Hole

A hole exists on the graph of a rational function at any input value that causes both the numerator and denominator of the function to be equal to zero.
limit

limit

A limit is the value that the output of a function approaches as the input of the function approaches a given value.