Define $f: R \rightarrow R$ by $ \left.f(x)=\cos \left(\tan ^{-1} \sin \left(\tan ^{-1}…

Define $f: R \rightarrow R$ by $ \left.f(x)=\cos \left(\tan ^{-1} \sin \left(\tan ^{-1} x\right)\right)\right) \text {, then } \lim _{x \rightarrow \infty}(f o f) x $ is equal to
  1. $\frac{3}{2 \sqrt{3}}$
  2. $\frac{\sqrt{2}}{3}$
  3. $\sqrt{\frac{2}{3}}$
  4. $\frac{2}{3 \sqrt{3}}$

Solution

Given, $f: R \rightarrow R$ by $ f(x)=\cos \left[\tan ^{-1}\left\{\sin \left(\tan ^{-1} x\right)\right\}\right] $ $ \text { To find, } \lim _{x \rightarrow \infty}(f o f)(x) $ Let us find ( $f \circ f)(x)$ first, $ \begin{aligned} & \therefore(f \circ f)(x)=f(f(x)) \\ & =f\left(\cos \left(\tan ^{-1}\left(\sin \left(\tan ^{-1} x\right)\right)\right)\right. \\ & =f\left(\cos \left(\tan ^{-1}\left(\sin \left(\sin ^{-1} \frac{x}{\sqrt{1+x^2}}\right)\right)\right)\right) \\ & \left\{\because \tan ^{-1} x=\sin ^{-1} \frac{x}{\sqrt{1+x^2}}\right\} \\ & =f\left(\cos \left(\tan ^{-1} \frac{x}{\sqrt{1+x^2}}\right)\right) \\ & \left\{\because \tan ^{-1} x=\cos ^{-1} \frac{1}{\sqrt{1+x^2}}\right\} \\ & =f\left(\cos \left(\cos ^{-1} \frac{1}{\sqrt{1+\left(\frac{x}{\sqrt{1+x^2}}\right)^2}}\right)\right) \\ & \end{aligned} $ $ \begin{aligned} & =f\left(\sqrt{\frac{1+x^2}{1+2 x^2}}\right) \\ \therefore(f \circ f)(x) & =\sqrt{\frac{1+\left(\sqrt{\frac{1+x^2}{1+2 x^2}}\right)^2}{1+2\left(\sqrt{\frac{1+x^2}{1+2 x^2}}\right)^2} \text { where, } f(x)=\sqrt{\frac{1+x^2}{1+2 x^2}}} \\ & =\sqrt{\frac{1+2 x^2+1+x^2}{1+2 x^2+2+2 x^2}}=\sqrt{\frac{2+3 x^2}{3+4 x^2}} \end{aligned} $ Let $x=1 / y$, then $y \rightarrow 0$, when $x \rightarrow \infty$ $ \begin{gathered} \therefore \lim _{x \rightarrow \infty} \sqrt{\frac{2+3 x^2}{3+4 x^2}}=\lim _{y \rightarrow 0} \sqrt{\frac{2+\frac{3}{y^2}}{3+\frac{4}{y^2}}} \\ =\lim _{y \rightarrow 0} \sqrt{\frac{2 y^2+3}{3 y^2+4}}=\sqrt{\frac{3}{4}} \\ =\frac{\sqrt{3}}{2} \text { or } \frac{3}{2 \sqrt{3}} \\ \therefore \quad \lim _{x \rightarrow \infty}(f \circ f)(x)=\frac{3}{2 \sqrt{3}} \end{gathered} $

Asked in: AP EAMCET 2021 (25 Aug Shift 2)

Practice more Functions questions on Aicharya