Question

Let $\left\{x\right\}$ denote the fractional part of $x$. Then $\underset{x\to 0}{lim}\frac{\left\{x\right\}}{\tan \left\{x\right\}}$

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

Answer 1

The correct option is D does not exist

Given, $\underset{x\to 0}{lim}\frac{\left\{x\right\}}{\tan \left\{x\right\}}$

L.H.L=$\underset{x\to 0^{-}}{lim}\frac{x-\left[x\right]}{\tan (x-[x\left]\right)}$

=$\underset{x\to 0^{-}}{lim}\frac{x+1}{\tan (x+1)}$

$=\frac{1}{\tan 1}=\cot 1$

$\therefore L.H.L=\cot 1$

$R.H.L=\underset{x\to 0^{+}}{lim}\frac{x-\left[x\right]}{\tan (x-[x\left]\right)}$

$\underset{x\to 0^{-1}}{lim}\frac{x}{\tan x}$

$=1$

$\therefore L.H.L\ne R.H.L$

Hence, Limit does not exist.