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 $\displaystyle \frac{1}{4}$.


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