Question

Let f : $r\to R$ he such that f (1) = 3 and f ' (1) = 6. Then,
$li{m}_{x\to 0}[\frac{f(1+x)}{f(1)}{]}^{1/x}$ equals

• A. 1
• B. $e^{\frac{1}{2}}$
• C. $e^2$
• D. $e^3$

Answer 1

Answer: (C)

Let y = $[\frac{f(1+x)}{f(1)}{]}^{1/x}\Rightarrow logy=\frac{1}{x}\{logf(1+x)-logf(1)]$
$\Rightarrow li{m}_{x\to 0}logy=li{m}_{x\to 0}[\frac{1}{f(1+x)}.f{}^{\prime }(1+x)]$
[using L' Hospital's rule]
$=\frac{f(1)}{f(1)}=\frac{6}{3}$
$\Rightarrow log\left(li{m}_{x\to 0}y\right)=2\Rightarrow li{m}_{x\to 0}y=e^2$