Single Variable Calculus, Chapter 2, 2.2, Section 2.2, Problem 21

Estimate the value of the $\displaystyle \lim \limits_{x \to 0} \frac{\sqrt{x + 4} - 2}{x}$ by using a table of values.



Let the values of $x$ be...

$
\begin{equation}
\begin{aligned}


\begin{array}{|c|c|}
\hline\\
x & f(x) \\
\hline\\
-0.1 & 0.251582 \\
-0.01 & 0.250156 \\
-0.001 & 0.250016 \\
-0.0001 & 0.250001 \\
-0.00001 & 0.25 \\
0.00001 & 0.249999 \\
0.0001 & 0.249998 \\
0.001 & 0.249984 \\
0.01 & 0.249844 \\
0.1 & 0.248457\\
\hline
\end{array}

\end{aligned}
\end{equation}
$



The table shows that as $x$ approaches 0 from both directions the
value of the limit approaches 0.25 or $\displaystyle \frac{1}{4}$.


$
\begin{equation}
\begin{aligned}

\displaystyle \lim \limits_{x \to 0} \frac{\sqrt{x + 4} - 2}{x} =& \frac{\sqrt{.000001 +4} - 2}{.000001} = \frac{1}{4}\\

\end{aligned}
\end{equation}
$