Evaluate the following limit: _ x 0 1- 5x 1- 6x - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{1-\cos5\text{x}}{1-\cos6\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{1-\cos5\text{x}}{1-\cos6\text{x}}$
$=\frac{\lim\limits_{\text{x}\rightarrow0}2\sin^2\frac{5\text{x}}{2}}{\lim\limits_{\text{x}\rightarrow0}2\sin^23\text{x}}$
$=\frac{2\Bigg(\lim\limits_{\text{x}\rightarrow0}\frac{\sin\frac{5\text{x}}{2}}{\frac{5\text{x}}{2}}\Bigg)^2\times\frac{25}{4}\text{x}^2}{2\Big(\lim\limits_{\text{x}\rightarrow0}\frac{\sin3\text{x}}{3\text{x}}\Big)^2\times9\text{x}^2}$
$=\frac{2\times1\times\frac{25}{4}\text{x}^2}{2\times1\times9\text{x}^9}$
$=\frac{25}{4\times9}$
$=\frac{25}{36}$

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