Q. $\lim_{x \rightarrow 1}\frac{\sqrt{1 - \cos{2(x - 1)}}}{x - 1}$
A) Exists and is equals $\sqrt{2}$
B) Exists and is equals $- \sqrt{2}$
C) Does not exist because $x - 1 \rightarrow 0$
D) Does not exist because left hand limit is not equal to right hand limit
Explanation
(d)
RHL$= \lim_{h \rightarrow 0}\ f(1 + h)$
$= \lim_{h \rightarrow 0}\ \frac{\sqrt{1 - \cos{2h}}}{h}$
$= \lim_{h \rightarrow 0}\ \ \frac{\sqrt{2}\sin h}{h} = \sqrt{2}$
LHL$= \lim_{h \rightarrow 0}\ \ f(1 - h)\lim_{h \rightarrow 0}\ \frac{\sqrt{1 - \cos{( - 2h)}}}{h}$
$= \lim_{h \rightarrow 0}\sqrt{2}\ \frac{\sin h}{- h} = - \sqrt{2}$
Here, LHL$\neq$RHL
So, limit does not exist.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.