Question

Let $f,g$ and $h$ be the real valued functions defined on $R$ as
$f(x)=\{\begin{array}{ll}\frac{x}{|x|},& x\ne 0\\ 1,& x=0\end{array},g(x)=\{\begin{array}{ll}\frac{\sin (x+1)}{(x+1)},& x\ne -1\\ 1,& x=-1\end{array}$
and $h(x)=2[x]-f(x)$, where $[x]$ is the greatest integer $\le x$.
Then the value of $\underset{x\to 1}{\lim }g(h(x-1))$ is

• A. 1
• B. 0
• C. $\sin (1)$
• D. $-1$

Answer 1

Answer: (A)

$LHL=\underset{k\to 0}{\lim }g(h(-k)),k>0$
$=\underset{k\to 0}{\lim }g(-2+1)$
$\because f(x)=-1\forall x<0$
$=g(-1)=1$
$RHL=\underset{k\to 0}{\lim }g(h(k)),k>0$
$=\underset{k\to 0}{\lim }g(-1),$
$\because f(x)=1,\forall x>0$
$=1$