Evaluate the given limit

by RoRi

September 3, 2015

Compute

$\lim_{x \to 0} \frac{1 - \sqrt{1-4x^2}}{x^2}.$

To simplify the expression we multiply the numerator and denominator by $1 + \sqrt{1-4x^2}$ to obtain,

$\begin{align*} \lim_{x \to 0} \frac{1 - \sqrt{1-4x^2}}{x^2} &= \lim_{x \to 0} \frac{(1-\sqrt{1-4x^2})(1+\sqrt{1-4x^2})}{x^2(1+\sqrt{1-4x^2}} \\ &= \lim_{x \to 0} \frac{1 - (1-4x^2)}{x^2(1+\sqrt{1-4x^2})} \\ &= \lim_{x \to 0} \frac{4}{1+\sqrt{1-4x^2}} \\ &= 2. \end{align*}$

One comment

August 3, 2022 at 1:44 pm

Jake says:

typo in the book solution.

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Stumbling Robot

Evaluate the given limit

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