Limits Question 4
Question 4 - 30 January - Shift 2
Let $f, g$ and $h$ be the real valued functions defined
on $\mathbb{R}$ as $f(x)=\begin{matrix} \frac{x}{|x|}, & x \neq 0 \\ 1, & x=0\end{matrix} .$,
$g(x)=\begin{matrix} \frac{\sin (x+1)}{(x+1)}, & x \neq-1 \\ 1, & x=-1\end{matrix} .$ and $h(x)=2[x]-f(x)$,
where $[x]$ is the greatest integer $\leq x$. Then the value of $\lim _{x \to 1} g(h(x-1))$ is
(1) 1
(2) $\sin (1)$
(3) -1
(4) 0
Show Answer
Answer: (1)
Solution:
Formula: Composition of functions, Algebra of limits
$LHL=\lim _{k \to 0} g(h(-k)) \quad, k>0$
$=\lim _{k \to 0} g(-2+1)$ $\because f(x)=-1 \forall x<0$
$=g(-1)=1$
RHL $=\lim _{k \to 0} g(h(k)) \quad, k>0$
$=\lim _{k \to 0} g(-1) \quad, \because f(x)=1, \forall x>0$
$=1$
