SATHEE: Limit Continuity and Differentiability 1 Question 11

Limit Continuity and Differentiability 1 Question 11

12. $lim_{h \to 0} \frac{f (2 h + 2 + h^{2}) - f (2)}{f (h - h^{2} + 1) - f (1)}$ , given that $f^{'} (2) = 6$ and $f^{'} (1) = 4$ ,

(2003, 2M)

(a) does not exist

(b) is equal to $- 3 / 2$

(c) is equal to $3 / 2$

(d) is equal to 3

Show Answer

Answer:

Correct Answer: 12. (d)

Solution:

Let $y = \frac{f (1 + x)^{1 / x}}{f (1)} \Rightarrow \log y = \frac{1}{x} [\log f (1 + x) - \log f (1)]$

$\Rightarrow lim_{x \to 0} \log y = lim_{x \to 0} \frac{1}{f (1 + x)} \cdot f^{'} (1 + x)$

$= \frac{f (1)}{f (1)} = \frac{6}{3}$

[using L’ Hospital’s rule]

$\Rightarrow \log lim_{x \to 0} y = 2 \Rightarrow lim_{x \to 0} y = e^{2}$

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