Single Variable Calculus, Chapter 2, Review Exercises, Section Review Exercises, Problem 15

Determine $\displaystyle \lim \limits_{x \to 0} \frac{1 - \sqrt{1 - x^2}}{x}$


$
\begin{equation}
\begin{aligned}

\lim \limits_{x \to 0} \frac{1 - \sqrt{1 - x^2}}{x} \cdot \frac{1 + \sqrt{1 - x^2}}{1 + \sqrt{1 - x^2}} &= \lim \limits_{x \to 0} \frac{1 - (1 - x^2)}{x(1 + \sqrt{1 - x^2})}
&& \text{Multiply numerator and denominator by $(1 + \sqrt{1 - x^2})$ then simplify}\\
\\
& = \lim \limits_{x \to 0} \frac{\cancel{(x)} (x)}{\cancel{x}(1 + \sqrt{1 - x^2})}
&& \text{Factor numerator and cancel out like terms}\\
\\
&= \frac{0}{1 + \sqrt{1 - (0)^2}} = \frac{0}{2}
&& \text{Substitute value of $x$ and simplify}\\
\\
& \fbox{$ = 0$}

\end{aligned}
\end{equation}
$