Question

If $f\left(x\right)=\{\begin{array}{cc}\left[tan\right(\frac{\pi }{4}+x){]}^{1/x}& x\ne 0\\ k& x=0\end{array}$ ,then for what value of $k$, $f\left(x\right)$ is continuous at $x=0$?

• A. $1$
• B. $e$
• C. $e^2$
• D. $e^{-1}$

Answer 1

The correct option is C $e^2$

$\underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0}{lim}\left[\tan \right(\pi /4+x){]}^{\frac{1}{x}}$
$=\underset{x\to 0}{lim}[\frac{1+\tan x}{1-\tan x}{]}^{\frac{1}{x}}$
$=\underset{x\to 0}{lim}(1+\tan x{)}^{\frac{1}{x}}(1-\tan x{)}^{-\frac{1}{x}}$
$=\underset{x\to 0}{lim}\left[\right(1+tanx{)}^{\frac{1}{tanx}}{]}^{\frac{tanx}{x}}\cdot \underset{x\to 0}{lim}\left[\right(1+(-tanx){)}^{\frac{1}{-tanx}}{]}^{\frac{tanx}{x}}$
$=e.e=e^2$
Given , $f\left(x\right)$ is continuous at $x=0$
$\Rightarrow k=e^2$