Question

Let $f, g$ and $h$ be the real valued functions defined on $R$ as
$f(x)=\begin{cases} \frac{x}{|x|}, & x \neq 0 \\1, & x=0\end{cases}, g(x)=\begin{cases} \frac{\sin (x+1)}{(x+1)}, & x \neq-1 \\1, & x=-1\end{cases}$
and $h(x)=2[x]-f(x)$, where $[x]$ is the greatest integer $\leq x$.
Then the value of $\displaystyle\lim _{x \rightarrow 1} g(h(x-1))$ is

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

Answer 1

Answer: (A)

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