Chapter 11.1, Problem 16E

To determine

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

Answer to Problem 16E

  $\underset{x\to -3}{\lim }f\left(x\right)=.25$

  

Explanation of Solution

Given:

  $\underset{x\to -3}{\lim }\frac{\sqrt{1-x}-2}{x+3}$

Calculation:

Use your graphing calculator to create a table of values of $f\left(x\right)$ for $x$ around $-4$ , and see if the function appears to approach a particular value as $x$ approaches $-4$ from both the left and right.

$x$	$f\left(x\right)$
$\begin{array}{l}-3.1\hfill \\ -3.01\hfill \\ -3.001\hfill \\ -3\hfill \\ -2.999\hfill \\ -2.99\hfill \\ -2.9\hfill \end{array}$	$\begin{array}{l}-0.248457\hfill \\ -0.249844\hfill \\ -0.249984\hfill \\ Indeterminate\hfill \\ -0.250016\hfill \\ -0.250156\hfill \\ -0.251582\hfill \end{array}$

Here, $f\left(x\right)$ takes values close to $-.25$ as $x$ takes values close to $-3$ , even though $f\left(x\right)$ is not defined there. Thus, $\underset{x\to -3}{\lim }f\left(x\right)=.25$ This can also be seen from the graph:

  

Conclusion:

Therefore, the value of the limit is $\underset{x\to -3}{\lim }f\left(x\right)=.25$