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Question: The value of \(\lim_{x \rightarrow 0}\frac{\sqrt{1 - x^{2}} - \sqrt{1 + x^{2}}}{x^{2}}\) is...

The value of lim⁡x→01−x2−1+x2x2\lim_{x \rightarrow 0}\frac{\sqrt{1 - x^{2}} - \sqrt{1 + x^{2}}}{x^{2}} is

A

1

B

–1

C

– 2

D

0

Answer

–1

Explanation

Solution

lim⁡x→0(1−x2−1+x2)x2(1−x2+1+x2)(1−x2+1+x2)\lim_{x \rightarrow 0}\frac{\left( \sqrt{1 - x^{2}} - \sqrt{1 + x^{2}} \right)}{x^{2}}\frac{\left( \sqrt{1 - x^{2}} + \sqrt{1 + x^{2}} \right)}{\left( \sqrt{1 - x^{2}} + \sqrt{1 + x^{2}} \right)}

= lim⁡x→0(1−x2)−(1+x2)x2(1−x2+1+x2)\lim_{x \rightarrow 0}\frac{(1 - x^{2}) - (1 + x^{2})}{x^{2}\left( \sqrt{1 - x^{2}} + \sqrt{1 + x^{2}} \right)} =−22=−1\frac{- 2}{2} = - 1.