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:

  1. 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]

$\Rightarrow \log \lim _{x \rightarrow 0} y=2 \Rightarrow \quad \lim _{x \rightarrow 0} y=e^{2}$



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