Establish the given limit formula

by RoRi

August 28, 2015

Show that

$\lim_{x \to 0} \frac{1 - \sqrt{1-x^2}}{x^2} = \frac{1}{2}.$

Proof. We multiply the numerator and denominator by $1 + \sqrt{1-x^2}$ ,

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

Stumbling Robot

Establish the given limit formula

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