Question

$\underset{x\to 0}{lim}\frac{e^{\left[|\sin x|\right]}}{[x+1]}$ , (where $\left[\right]$ denotes the greatest integer)

• A. 1
• B. $\pi$
• C. does not exist
• D. $\frac{\pi }{2}$

Answer 1

The correct option is C does not exist

L.H.L. $=\underset{x\to 0^{-}}{lim}\frac{e^{\left[|\sin x|\right]}}{[x+1]}=\frac{e^0}{\left[1^{-}\right]}=\frac{1}{0}=\infty$
R.H.L.$=\underset{x\to 0^{+}}{lim}\frac{e^{\left[|\sin x|\right]}}{[x+1]}=\frac{e^0}{\left[1^{+}\right]}=\frac{1}{1}=1$
Clearly L.H.L.$\ne$ R.H.L.$\Rightarrow$ limit does not exist