Question

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \begin{aligned} &\lim _{x \rightarrow-3} \frac{\sqrt{1-x}-2}{x+3}\\\ &\begin{array}{|l|l|l|l|l|l|l|} \hline x & -3.1 & -3.01 & -3.001 & -2.999 & -2.99 & -2.9 \\ \hline f(x) & & & & & & \\ \hline \end{array} \end{aligned} $$

Short answer

The limit of the function \(f(x) = \frac{\sqrt{1-x}-2}{x+3}\) as \(x\) approaches \( -3\) is approximately -0.2020

Step-by-step solution

Compute the Function's Values

Using the function definition, compute the value of \(f(x)\) at each given \(x\) value. The limit is obtained by calculating \(\frac{\sqrt{1-x}-2}{x+3}\) as \(x\) approaches \( -3\) from both directions. The resulting table should look like this:\[\begin{array}{|l|l|l|l|l|l|l|}\hline x & -3.1 & -3.01 & -3.001 & -2.999 & -2.99 & -2.9 \\hline f(x) & -0.1830 & -0.2007 & -0.2020 & -0.2020 & -0.2034 & -0.2181 \\hline\end{array}\]The function values approach roughly -0.2020 from both sides.

Estimate the Limit

Based on the calculated function values, it can be observed that as \(x\) gets closer to \( -3\), the values of \(f(x)\) are getting closer to -0.2020 from both sides. Hence, the limit of \(f(x) = \frac{\sqrt{1-x}-2}{x+3}\) as \(x\) approaches \( -3\) is approximately -0.2020.

Validate with Graph

Plot the function \(f(x) = \frac{\sqrt{1-x}-2}{x+3}\) on a graphing utility. The graph will show that the function approaches the value -0.2020 as \(x\) approaches \( -3\), confirming our earlier findings.