Calculus and Its Applications, Chapter 1, Review Exercises, Section Review Exercises, Problem 54

Create an input-output table that includes the limit $\displaystyle \lim_{x \to 1} \frac{2 - \sqrt{x + 3}}{x - 1}$.
Start with $\Delta \text{Tbl} = 0.1$ and then go to $0.01, 0.001$ and $0.0001$. When you think you know the
limit, illustrate the function and use the TRACE feature to verify the answer.
With $\Delta \text{Tbl} = 0.1$

$
\begin{array}{|c|c|}
\hline
x & \displaystyle \frac{2 - \sqrt{x + 3}}{x - 1} \\
\hline
0.7 & -0.254 \\
\hline
0.8 & -0.253 \\
\hline
0.9 & -0.251 \\
\hline
1.0 & \text{ERROR} \\
\hline
1.1 & -0.248 \\
\hline
1.2 & -0.246 \\
\hline
1.3 & -0.245 \\
\hline
\end{array}
$


With $\Delta \text{Tbl} - 0.01$

$
\begin{array}{|c|c|}
\hline
x & \displaystyle \frac{2 - \sqrt{x + 3}}{x - 1} \\
\hline
0.97 & -0.25 \\
\hline
0.98 & -0.25 \\
\hline
0.99 & -0.25 \\
\hline
1.0 & \text{ERROR} \\
\hline
1.01 & -0.249 \\
\hline
1.02 & -0.249 \\
\hline
1.03 & -0.249 \\
\hline
\end{array}
$


Based on the values from the table, it seems that the
$\displaystyle \lim_{x \to 1} = -0.25 \text{ or } -\frac{1}{4}$

Then, by graphing $\displaystyle f(x) = \frac{2 - \sqrt{x + 3 }}{x - 1}$, we have


We can see from the graph that the $\displaystyle \lim_{x \to 1} f(x) = -0.25$