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$} \end{aligned} \end{equation}$