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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{1-\cos\text{mx}}{\text{x}^2}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\sin^2\frac{\text{mx}}{2}}{\text{x}^2}$
$=2\times\bigg(\lim\limits_{\text{x}\rightarrow0}\frac{\sin\frac{\text{mx}}{2}}{\text{x}}\bigg)^2$
$=2\times\bigg(\lim\limits_{\text{x}\rightarrow0}\frac{\sin\frac{\text{mx}}{2}}{\frac{\text{mx}}{2}}\bigg)^2\times\Big(\frac{\text{m}}{2}\Big)$
$=2\times\frac{\text{m}^2}{4}=\frac{\text{m}^2}{2}$ $\Big[\because\ \lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}=1\Big]$
$=\frac{\text{m}^2}{2}$

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