Question

The value of $\underset{x\to 0}{lim}{\left(\frac{1+5x^2}{1+3x^2}\right)}^{1/x^2}=$

• A. $e$
• B. $e^2$
• C. $1/e$
• D. $0$

Answer 1

Answer: (B)

$e^2$

On applying the limit, since, the expression is of the form $1^{\infty }$,
we rewrite it in the form of,
${lim}_x\to 0exp\left[\right(\frac{1+5x^2}{1+3x^2}-1)\times \frac{1}{x^2}]$

$={lim}_x\to 0exp\left[\right(\frac{2x^2}{1+3x^2}\times \frac{1}{x^2}]$

$={lim}_x\to 0exp\left[\frac{2}{1+3x^2}\right]$

$=e^2$