is equal to :

Engineering

Mathematics

One to The Power Infinity Form

Question

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

e²

$\frac{1}{e}$

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

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