Q55 / 125 · BITSAT 2010 · Mathematics

Find the value of \(\lim _{x \rightarrow 0} \frac{\sqrt{\left(1+x^2\right)}-\sqrt{1-x^2}}{x^2}\)

  • Correct answer
    A
    1
  • B
    2
  • C
    3
  • D
    5

Solution

\(\begin{aligned} & \lim _{x \rightarrow 0} \frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{x^2} \cdot \frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}} \\ \Rightarrow & \lim _{x \rightarrow 0} \frac{1+x^2-1+x^2}{x^2\left(\sqrt{1+x^2}+\sqrt{1-x^2}\right)} \\ \Rightarrow & \lim _{x \rightarrow 0} \frac{2 x^2}{x^2\left(\sqrt{1+x^2}+\sqrt{1-x^2}\right)}=\frac{2}{\sqrt{1}+\sqrt{1}}=\frac{2}{2}=1 \end{aligned}\)

Hence, the correct answer is option: 'A'.

More Statistics Questions