# What is the best method to solve the limit \lim_{x\to

What is the best method to solve the limit $\underset{x\to \mathrm{\infty }}{lim}{\left(1+\mathrm{sin}\frac{2}{{x}^{2}}\right)}^{{x}^{2}}$?
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Hint:
$\underset{x\to \mathrm{\infty }}{lim}{\left(1+\mathrm{sin}\frac{2}{{x}^{2}}\right)}^{{x}^{2}}={\left(\underset{x\to \mathrm{\infty }}{lim}{\left(1+\mathrm{sin}\frac{2}{{x}^{2}}\right)}^{\frac{1}{\mathrm{sin}\left(\frac{2}{{x}^{2}}\right)}}\right)}^{2\underset{x\to \mathrm{\infty }}{lim}\frac{\mathrm{sin}\frac{2}{{x}^{2}}}{\frac{2}{{x}^{2}}}}$
Now for the exponent set $\frac{2}{{x}^{2}}=h⇒h\to {0}^{+}$
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chabinka61jx
For questions like this one, I've found asymptotic calculations to be handy. For example:
${\left(1+\mathrm{sin}\frac{2}{{x}^{2}}\right)}^{{x}^{2}}={e}^{{x}^{2}\mathrm{ln}\left(1+\mathrm{sin}\frac{2}{{x}^{2}}\right)}={e}^{{x}^{2}\left(\mathrm{sin}\frac{2}{{x}^{2}}+o\left(\mathrm{sin}\frac{2}{{x}^{2}}\right)\right)}={e}^{{x}^{2}\left(\frac{2}{{x}^{2}}+o\left(\frac{2}{{x}^{2}}\right)+o\left(\frac{2}{{x}^{2}}\right)\right)}={e}^{2+o\left(1\right)}\to {e}^{2}\left(x\to \mathrm{\infty }\right)$