Question

Let $f:R\to R$ be such that $f\left(a\right)=1,f^{\prime }\left(a\right)=2$. Then $\underset{x\to 0}{lim}{\left(\frac{f^2(a+x)}{f\left(a\right)}\right)}^{1/x}$ is

• A. $e^2$
• B. $e^4$
• C. $e^{-4}$
• D. $1/e$

Answer 1

The correct option is A $e^2$

${lim}_{x\to 0}{\left(\frac{f^2(a+x)}{f\left(a\right)}\right)}^{\frac{1}{x}}$

$=e^{{lim}_{x\to 0}\frac{1}{x}\left[\log f^2\right(a+x)-\log f(a\left)\right]}$

$=e^{{lim}_{x\to 0}\frac{\frac{1}{f(a+x)f^{\prime }(a+x)}}{2}}$

$=e^{\frac{f^{\prime }\left(1\right)}{f\left(1\right)}}$

$=e^{\frac{2}{1}}$

$=e^2$