Question:
$\lim _{x \rightarrow 0}\left(\frac{3 x^{2}+2}{7 x^{2}+2}\right)^{1 / x^{2}}$ is equal to:
Correct Option: , 2
Solution:
Let $\mathrm{R}=\lim _{x \rightarrow 0}\left(\frac{3 x^{2}+2}{7 x^{2}+2}\right)^{\frac{1}{x^{2}}}=e^{\lim _{x \rightarrow 0} \frac{1}{x^{2}}\left\{\frac{3 x^{2}+2}{7 x^{2}+2}-1\right\}}$
$=e^{\lim _{x \rightarrow 0} \frac{1}{x^{2}}\left\{\frac{-4 x^{2}}{7 x^{2}+2}\right\}}=e^{\frac{-4}{2}}=e^{-2}=\frac{1}{e^{2}}$