Evaluate the following limit: _ x 0 ^24x^2 x^4 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sin^24\text{x}^2}{\text{x}^4}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sin^24\text{x}^2}{\text{x}^4}$

$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\sin4{\text{x}^2\big)}^{2}}{\text{x}^4}$

$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\sin4\text{x}^2\big)^2}{\big(\text{x}^2\big)^2}$

$=\Big(\lim\limits_{\text{x}\rightarrow0}\frac{\sin4\text{x}^2}{\text{x}^2}\Big)^2$

$=\Big(\lim\limits_{4\text{x}^2\rightarrow0}\frac{\sin4\text{x}^2}{4\text{x}^2}\Big)\times16$

$=1\times16$ $\Big[\because\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}=1\Big]$

$=16$

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