Question

If $\left\{x\right\}$ denotes the fractional part of $x,$ then value of
$\underset{x\to 0}{lim}\frac{\left\{x\right\}}{\tan \left\{x\right\}}$ is

• A. $1$
• B. $0$
• C. $-1$
• D. It does not exist

Answer 1

The correct option is D It does not exist

$\underset{x\to 0}{lim}\frac{\left\{x\right\}}{\tan \left\{x\right\}}$
We know, for $x\to 0^{+},\left\{x\right\}=x\Rightarrow \left\{x\right\}\to 0$
But for $x\to 0^{-},\left\{x\right\}=1+x\Rightarrow \left\{x\right\}\to 1$

$LHL=\underset{x\to 0^{-}}{lim}\frac{\left\{x\right\}}{\tan \left\{x\right\}}=\frac{1}{\tan 1}$
$RHL=\underset{x\to 0^{+}}{lim}\frac{x}{\tan x}=1$
Clearly, $LHL\ne RHL$
Hence, limit does not exist