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

### Using Graphs to Find Limits

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.

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At **neighborhood** around 0. By neighborhood around 0, we mean what is happening on the lines around

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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.

### Examples

#### Example 1

Earlier, you were asked to find

#### Example 2

Evaluate the following expressions using the graph of the function

limx→−∞f(x) limx→−1f(x) limx→0f(x) limx→1f(x) limx→3f(x) f(−1) f(2) f(1) f(3)

limx→−∞f(x)=0 limx→−1f(x)=DNE limx→0f(x)=−2 limx→1f(x)=0 limx→3f(x)=DNE (This is because only one side exists and a regular limit requires both left and right sides to agree)f(−1)=0 f(0)=−2 f(1)=2 f(3)=0

#### Example 3

Sketch a graph that has a limit at

While there are an infinite number of graphs that fit this criteria, you should make sure your graph has a removable discontinuity at

#### Example 4

Sketch a graph that is defined at

The graph must have either a jump or an infinite discontinuity at

#### Example 5

Evaluate and explain how to find the limits as \begin{align*}x\end{align*} approaches 0 and 1 for the graph below:

\begin{align*}\lim_{x \to 0}f(x)=2, \ \lim_{x \to 1}f(x)=1\end{align*}

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 \begin{align*}(0,2)\end{align*} the function happened to be defined there. In the case of the point \begin{align*}(1,1)\end{align*} 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.

### Review

Use the graph of \begin{align*}f(x)\end{align*} below to evaluate the expressions in 1-6.

\begin{align*}\lim_{x \to - \infty}f(x)\end{align*}1.

\begin{align*}\lim_{x \to \infty}f(x)\end{align*}2.

\begin{align*}\lim_{x \to 2}f(x)\end{align*}3.

\begin{align*}\lim_{x \to 0}f(x)\end{align*}4.

\begin{align*}f(0)\end{align*}5.

\begin{align*}f(2)\end{align*}6.

\begin{align*}g(x)\end{align*} below to evaluate the expressions in 7-13.Use the graph of

7. \begin{align*}\lim_{x \to - \infty}g(x)\end{align*}

8. \begin{align*}\lim_{x \to \infty}g(x)\end{align*}

9. \begin{align*}\lim_{x \to 2}g(x)\end{align*}

10. \begin{align*}\lim_{x \to 0}g(x)\end{align*}

11. \begin{align*}\lim_{x \to 4}g(x)\end{align*}

12. \begin{align*}g(0)\end{align*}

13. \begin{align*}g(2)\end{align*}

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

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

### Review (Answers)

To see the Review answers, open this PDF file and look for section 14.2.