Chapter 11.1, Problem 19E

To determine

To create: a table,estimate the limit and graph the function

Answer to Problem 19E

  $\underset{x\to 0}{\lim }f\left(x\right)=1$

  

Explanation of Solution

Given:

  $\underset{x\to 0}{\lim }\frac{\sin x}{x}$

Calculation:

Use your graphing calculator to create a table of values of $f\left(x\right)$ for $x$ around $0$ , and see if the function appears to approach a particular value as $x$ approaches $0$ from both the left and right.

$x$	$f\left(x\right)$
$\begin{array}{l}-0.1\hfill \\ -0.01\hfill \\ -0.001\hfill \\ 0\hfill \\ 0.001\hfill \\ 0.01\hfill \\ 0.1\hfill \end{array}$	$\begin{array}{l}0.998334\hfill \\ 0.999983\hfill \\ 1\hfill \\ Indeterminate\hfill \\ 1\hfill \\ 0.999983\hfill \\ 0.998334\hfill \end{array}$

Here, $f\left(x\right)$ takes values close to $1$ as $x$ takes values close to $0$ , even though $f\left(x\right)$ is not defined there. Thus, $\underset{x\to 0}{\lim }f\left(x\right)=1$ . This can also be seen from the graph:

  

Conclusion:

Therefore, the value of the limit is $\underset{x\to 0}{\lim }f\left(x\right)=1$