Differentiation Question 189
Question: Let $ f:R\to R $ be such that $ f(1)=3 $ and $ f’(1)=6. $ Then $ \underset{x\to 0}{\mathop{\lim }}{{( \frac{f(1+x)}{f(1)} )}^{1/x}} $ equals
Options:
A) 1
B) $ {e^{1/2}} $
C) $ e^{2} $
D) $ e^{3} $
Show Answer
Answer:
Correct Answer: C
Solution:
[c] Given that $ f:R\to R $ such that $ f(1)=3andf’(1)=6 $ Then $ \underset{x\to 0}{\mathop{\lim }}{{[ \frac{f(1+x)}{f(1)} ]}^{1/x}} $
$ ={e^{\underset{x\to 0}{\mathop{\lim }}\frac{1}{x}[logf(1+x)-logf(1)]}} $
$ ={e^{\underset{x\to 0}{\mathop{\lim }}\frac{\frac{1}{f(1+x)}f’(1+x)}{1}}}={e^{\frac{f’(1)}{f(1)}}}={e^{6/3}}=e^{2} $ [Using L Hospital rule]
BETA
