Single Variable Calculus, Chapter 2, 2.2, Section 2.2, Problem 42

(a) Estimate the value of the $\displaystyle \lim \limits_{x \to 1} \frac{x^3 - 1}{\sqrt{x} - 1}$ by using numerical and graphical evidence.



Based from the graph, the estimated valueof the $\displaystyle \lim \limits_{x \to 1} \frac{x^3 - 1}{\sqrt{x} - 1} \approx 6$

We let the values of $x$ be..


$
\begin{equation}
\begin{aligned}

\begin{array}{|c|c|}
\hline\\
x & \lim \limits_{x \to 1} \\
\hline\\
0.9 & 5.2809 \\
0.93 & 5.4902 \\
0.94 & 5.5612 \\
0.95 & 5.6328 \\
0.99 & 5.9253 \\
0.999 & 5.9925 \\
0.9999 & 5.9993 \\
0.99999 & 5.9999\\
\hline
\end{array}

\end{aligned}
\end{equation}
$


Based on the numerical values we obtain from the table, it seems that the $\displaystyle \lim \limits_{x \to 1} \frac{x^3 - 1}{\sqrt{x} - 1} = 6$

b.) Determine how close to 1 does $x$ have to be to ensure that the function is within a distance of 0.5 of its limit.

In order to ensure that the function in part (a) is within the distance $0.5$ of its limit, the values of
$x$ should be atleast $0.06$ closer to $1$ based on the values we obtain from the table.