Question

If $f\left(x\right)=e^{\left[\cot x\right]}$ where $\left[y\right]$ represents the greatest integer less than or equal to $y$, then

• A. $\underset{x\to \frac{{\pi }^{+}}{2}}{lim}f\left(x\right)=1$
• B. $\underset{x\to \frac{{\pi }^{+}}{2}}{lim}f\left(x\right)=\frac{1}{e}$
• C. $\underset{x\to \frac{{\pi }^{-}}{2}}{lim}f\left(x\right)=\frac{1}{e}$
• D. $\underset{x\to \frac{{\pi }^{-}}{2}}{lim}f\left(x\right)=1$

Answer 1

Answer: (B), (D)

$\underset{x\to \frac{{\pi }^{+}}{2}}{lim}f\left(x\right)=\frac{1}{e}$
$\underset{x\to \frac{{\pi }^{-}}{2}}{lim}f\left(x\right)=1$

$\cot x$ is decreasing function in $[0,\pi ]$

at $x=\pi /2\cot x=0$

as $x={\frac{\pi }{2}}^{-}\cot x>0$ , as $x={\frac{\pi }{2}}^{+}\cot x<0$

$\Rightarrow \left[\cot x\right]=0,\Rightarrow \left[\cot x\right]=-1$

$\underset{x\to {\pi /2}^{-}}{lim}f\left(x\right)=e^0=1$

$\underset{x\to {\pi /2}^{+}}{lim}f\left(x\right)=e^{-1}=\frac{1}{e}$