Evaluate the given limit

by RoRi

September 3, 2015

Compute

$\lim_{x \to 0} \frac{\sqrt{1+x} - \sqrt{1-x}}{x}.$

Here, we use the trick of multiplying the numerator and denominator by something convenient to get an expression we can evaluate

$\begin{align*} \lim_{x \to 0} \frac{\sqrt{1+x} - \sqrt{1-x}}{x} &= \lim_{x \to 0} \frac{(\sqrt{1+x} - \sqrt{1-x})(\sqrt{1+x} + \sqrt{1-x})}{x (\sqrt{1+x} + \sqrt{1-x})} \\ &= \lim_{x \to 0} \frac{(1+x)-(1-x)}{x (\sqrt{1+x} + \sqrt{1-x})} \\ &= \lim_{x \to 0} \frac{2}{\sqrt{1+x} + \sqrt{1-x}} \\ &= 1. \end{align*}$

Stumbling Robot

Evaluate the given limit

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