First slide

Introduction to limits

Question

$lim_{x \to 0} {(\frac{3 x^{2} + 2}{7 x^{2} + 2})}^{1 / x^{2}}$ is equal to

Moderate

A

$e^{- 2}$
B

$e^{2}$
C

$e^{2 / 7}$
D

$e^{3 / 7}$

Solution

$lim_{x \to 0} {(\frac{3 x^{2} + 2}{7 x^{2} + 2})}^{1 / x^{2}}$
$= lim_{x \to 0} {(1 + \frac{3 x^{2} + 2}{7 x^{2} + 2} - 1)}^{1 / x^{2}}$
$= lim_{x \to 0} {(1 + \frac{- 4 x^{2}}{7 x^{2} + 2})}^{1 / x^{2}} = e^{lim_{x \to 0} \frac{- 4 x^{2}}{7 x^{2} + 2} \frac{1}{x^{2}}} = e^{lim_{x \to 0} \frac{- 4}{7 x^{2} + 2}} = e^{- 2}$

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