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Evaluate the given limit

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*}

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