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)

1

Answer 2

The correct answer is (A) : 1
LHL=k→0lim​g(h(−k)),k>0
=k→0lim​g(−2+1)
∵f(x)=−1∀x<0
=g(−1)=1
RHL=k→0lim​g(h(k)),k>0
=k→0lim​g(−1),
∵f(x)=1,∀x>0
=1