Evaluate the following limit: _ x 0 1- 4x x^2 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{1-\cos4\text{x}}{\text{x}^2}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{1-\cos4\text{x}}{\text{x}^2}$
$=\lim\limits_{\text{x} \rightarrow0}\frac{2\sin^22\text{x}}{\text{x}^2}$
$=2\lim\limits_{\text{x} \rightarrow0}\Big(\frac{\sin2\text{x}}{\text{x}}\Big)^2$
$=2\lim\limits_{\text{x}\rightarrow0}\Big(\frac{\sin2\text{x}}{2\text{x}}\Big)^2\times(2)^2$
$=2\times1\times4$
$=8$

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