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Q. Let $g(x)=\frac{(x-1)^{n}}{\log \cos ^{m}(x-1)} ; 0 < x < 2, m$ and $n$ are integers, $m \neq 0, n > 0$, and let $p$ be the left hand derivative of $|x-1|$ at $x=1$. If $\displaystyle\lim _{ s \rightarrow 1^{+}} g(x)=p$, then

JEE AdvancedJEE Advanced 2008

Solution:

image
From graph, $p=-1$
$\Rightarrow \displaystyle\lim _{ s \rightarrow 1^{+}} g(x)=-1$
$\Rightarrow \displaystyle\lim _{h \rightarrow 0} g(1+h)=-1$
$\Rightarrow \displaystyle\lim _{h \rightarrow 0}\left(\frac{h^{n}}{\log \cos ^{m} h}\right)=-1$
$\Rightarrow \displaystyle\lim _{h \rightarrow 0} \frac{n \cdot h^{n-1}}{m \cdot(-\tanh )}=-\left(\frac{n}{m}\right) \lim _{h \rightarrow 0}\left(\frac{h^{n-1}}{\tanh }\right)=-1$,
which holds if $n=m=2 .$