Limit Continuity and Differentiability 1 Question 11
12. $\lim _{h \rightarrow 0} \frac{f\left(2 h+2+h^{2}\right)-f(2)}{f\left(h-h^{2}+1\right)-f(1)}$, given that $f^{\prime}(2)=6$ and $f^{\prime}(1)=4$,
(2003, 2M)
(a) does not exist
(b) is equal to $-3 / 2$
(c) is equal to $3 / 2$
(d) is equal to 3
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Answer:
Correct Answer: 12. (d)
Solution:
- Let $y=\frac{f(1+x)^{1 / x}}{f(1)} \Rightarrow \log y=\frac{1}{x}[\log f(1+x)-\log f(1)]$
$\Rightarrow \quad \lim _{x \rightarrow 0} \log y=\lim _{x \rightarrow 0} \frac{1}{f(1+x)} \cdot f^{\prime}(1+x)$
$$ =\frac{f(1)}{f(1)}=\frac{6}{3} $$
[using L’ Hospital’s rule]
$\Rightarrow \log \lim _{x \rightarrow 0} y=2 \Rightarrow \quad \lim _{x \rightarrow 0} y=e^{2}$
