Evaluate the following limits:

Question:

Evaluate the following limits:.

$\lim _{x \rightarrow 0} \frac{(1-\cos 2 x)}{(\cos 2 x-\cos 8 x)}$ 

Solution:

$=\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{(\cos 2 x-\cos 8 x)}$

$=\lim _{x \rightarrow 0} \frac{2 \times \sin x \times \sin x}{2 \times \sin 3 x \times \sin 5 x} \times \frac{5 x \times 3 x}{x \times x} \times \frac{1}{15}$

$=\frac{1}{15} \times 1 \times 1 \times 1 \times 1\left[\because \lim _{x \rightarrow 0} \frac{\sin \theta}{\theta}=1\right]$

$=\frac{1}{15}$

$\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{(\cos 2 x-\cos 8 x)}=\frac{1}{15}$

 

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