1. Tardigrade
  2. Question
  3. Mathematics
  4. displaystyle lim x arrow 0 (1/x) cos -1((1-x2/1+x2)) is equal to

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. $\displaystyle\lim _{x \rightarrow 0} \frac{1}{x} \cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)$ is equal to

Limits and Derivatives

Solution:

We know that $\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)=\begin{cases}2 \tan ^{-1} x, & x \geq 0 \\ -2 \tan ^{-1} x, & x \leq 0\end{cases}$
or $\displaystyle\lim _{x \rightarrow 0^{+}} \frac{1}{x} \cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)$
$=\displaystyle\lim _{x \rightarrow 0^{+}} \frac{2 \tan ^{-1} x}{x}=2$
and $\displaystyle\lim _{x \rightarrow 0^{-}} \frac{1}{x} \cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)$
$=\displaystyle\lim _{x \rightarrow 0^{+}}\left[-\frac{2 \tan ^{-1} x}{x}\right]=-2$