Question

Let f : R → R be such that f(1) = 3 and f’(1) = 6. Then $\lim_{x \rightarrow 0}\left( \frac{f(1 + x)}{f(1)} \right)^{1/x}$ . Equals

• A. 1
• B. e^1/2
• C. e²
• D. e³

Answer 1

Answer: (C)

e²

Answer 2

Given that f; R → R s.t.

F(1) = 3 and f’(1) = 6

Then $\lim_{x \rightarrow 0}\left\lbrack \frac{f(1 + x)}{f(1)} \right\rbrack^{1/x}$= $\lim_{e^{x \rightarrow 0}}\frac{1}{x}\left\lbrack \log f(1 + x) - \log f(1) \right\rbrack$

= $\lim_{e^{x \rightarrow 0}}\frac{\frac{1}{f(1 + x)}f'(1 + x)}{1}$ [Using L’ Hospital rule]

= $e^{\frac{f'(1)}{f(1)}} = e^{6/3} = e^{2}$.