First slide

Introduction to limits

Question

The value of $lim_{x \to 0} {(\frac{1 + 5 x^{2}}{1 + 3 x^{2}})}^{1 / x^{2}}$ , is

Moderate

A

$e^{2}$
B

$e$
C

$e^{- 1}$
D

none of these

Solution

We have,
$lim_{x \to 0} {(\frac{1 + 5 x^{2}}{1 + 3 x^{2}})}^{\frac{1}{x^{2}}} = e^{lim_{x \to 0} \frac{2 x^{2}}{1 + 3 x^{2}} \cdot \frac{1}{x^{2}}} = e^{{lim_{x \to 0} \frac{2}{1 + 3 x^{2}}}_{}} = e^{2}$ .

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