Limit Continuity and Differentiability 7 Question 14
14. Let $g(x)=\frac{(x-1)^{n}}{\log \cos ^{m}(x-1)} ; 0<x<2, \quad m \quad$ and $n$ are integers, $m \neq 0, n>0$ and let $p$ be the left hand derivative of $|x-1|$ at $x=1$. If $\lim _{x \rightarrow 1^{+}} g(x)=p$, then
(a) $n=1, m=1$
(b) $n=1, m=-1$
(c) $n=2, m=2$
(d) $n>2, m=n$
(2008, 3M)
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Answer:
Correct Answer: 14. (a, b, c, d)
Solution:
- Since, $\quad \frac{d x}{d y}=\frac{1}{d y / d x}=\frac{d y}{d x}^{-1}$
$$ \begin{aligned} \Rightarrow & \frac{d}{d y} \frac{d x}{d y} & =\frac{d}{d x} \frac{d y}{d x}{ }^{-1} \frac{d x}{d y} \\ \Rightarrow & \frac{d^{2} x}{d y^{2}} & =-{\frac{d^{2} y}{d x^{2}} \quad \frac{d y}{d x}}^{-2} \quad \frac{d x}{d y}=-{\frac{d^{2} y}{d x^{2}} \quad \frac{d y}{d x}}^{-3} \end{aligned} $$
