Question

$\underset{x\to 0}{lim}{\left(\frac{3x^2+2}{7x^2+2}\right)}^{1/x^2}$ is equal to:

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

Answer 1

The correct option is B $\frac{1}{e^2}$

Let $L=\underset{x\to 0}{lim}{\left(\frac{3x^2+2}{7x^2+2}\right)}^{1/x^2}\left[\text{Indeterminate form}{1}^{\infty }\right]$
$\therefore L=e^{\underset{x\to 0}{lim}}\frac{1}{x^2}(\frac{3x^2+2}{7x^2+2}-1)$
$=e^{\underset{x\to 0}{lim}}\frac{1}{x^2}\left(\frac{-4x^2}{7x^2+2}\right)$
$=\frac{1}{e^2}$