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 $\displaystyle \lim_{x \to 1} f(x) = -0.25$