Question

Let $f\left(x\right)=\frac{\tan \left({\displaystyle \frac{\mathrm{\pi }}{4}}-x\right)}{\cot 2x},x\ne \frac{\mathrm{\pi }}{4}.$ The value which should be assigned to f (x) at $x=\frac{\mathrm{\pi }}{4},$ so that it is continuous everywhere is
(a) 1
(b) 1/2
(c) 2
(d) none of these

Answer 1

(b) $\frac{1}{2}$

If $f\left(x\right)$ is continuous at $x=\frac{\mathrm{\pi }}{4}$, then
$\underset{\mathrm{x}\to \frac{\mathrm{\pi }}{4}}{\lim }f\left(x\right)=f\left(\frac{\mathrm{\pi }}{4}\right)$

$\Rightarrow \underset{\mathrm{x}\to \frac{\mathrm{\pi }}{4}}{\lim }\frac{\tan \left({\displaystyle \frac{\mathrm{\pi }}{4}}-x\right)}{\cot 2x}=f\left(\frac{\mathrm{\pi }}{4}\right)$


If $\frac{\mathrm{\pi }}{4}-x=y$, then $x\mathit{\to }\frac{\pi }{4}\mathit{}\mathrm{and}\mathit{}y\to 0$.

$$\begin{gathered} \therefore \underset{y\to 0}{\lim }\left(\frac{\tan y}{\cot 2\left({\displaystyle \frac{\mathrm{\pi }}{4}}-y\right)}\right)=f\left(\frac{\mathrm{\pi }}{4}\right) \\ \Rightarrow \underset{y\to 0}{\lim }\left(\frac{\tan y}{\cot \left({\displaystyle \frac{\mathrm{\pi }}{2}}-2y\right)}\right)=f\left(\frac{\mathrm{\pi }}{4}\right) \\ \Rightarrow \underset{y\to 0}{\lim }\left(\frac{\tan y}{\tan 2y}\right)=f\left(\frac{\mathrm{\pi }}{4}\right) \\ \Rightarrow \underset{y\to 0}{\lim }\left(\frac{{\displaystyle \frac{\tan y}{y}}}{{\displaystyle \frac{\tan 2y}{y}}}\right)=f\left(\frac{\mathrm{\pi }}{4}\right) \\ \Rightarrow \underset{y\to 0}{\lim }\left(\frac{{\displaystyle \frac{\tan y}{y}}}{{\displaystyle \frac{2\tan 2y}{2y}}}\right)=f\left(\frac{\mathrm{\pi }}{4}\right) \\ \Rightarrow \frac{1}{2}\underset{y\to 0}{\lim }\left(\frac{{\displaystyle \frac{\tan y}{y}}}{{\displaystyle \frac{\tan 2y}{2y}}}\right)=f\left(\frac{\mathrm{\pi }}{4}\right) \\ \Rightarrow \frac{1}{2}\left(\frac{\underset{y\to 0}{\lim }{\displaystyle \frac{\tan y}{y}}}{\underset{y\to 0}{\lim }{\displaystyle \frac{\tan 2y}{2y}}}\right)=f\left(\frac{\mathrm{\pi }}{4}\right) \\ \Rightarrow \frac{1}{2}\left(\frac{1}{1}\right)=f\left(\frac{\mathrm{\pi }}{4}\right) \\ \Rightarrow f\left(\frac{\mathrm{\pi }}{4}\right)=\frac{1}{2} \end{gathered}$$