Evaluate the following limits:
Question:

Evaluate the following limits:

$\lim _{x \rightarrow 0} \frac{(\cos 3 x-\cos 5 x)}{x^{2}}$

 

Solution:

$=\lim _{x \rightarrow 0} \frac{(1-\cos 5 x-(1-\cos 3 x))}{x^{2}}$

$=\lim _{x \rightarrow 0}\left(\frac{1-\cos 5 x}{x^{2}}-\frac{1-\cos 3 x}{x^{2}}\right)$

$=\left(\lim _{x \rightarrow 0} \frac{1-\cos 5 x}{x^{2}} \times \frac{25}{25}\right)-\left(\lim _{x \rightarrow 0} \frac{1-\cos 3 x}{x^{2}} \times \frac{9}{9}\right)$

$=\frac{25}{2}-\frac{9}{2}\left[\because \lim _{x \rightarrow 0} \frac{1-\cos a x}{(\operatorname{ax})^{2}}=\frac{1}{2}\right]$

$=\frac{16}{2}$

$=8$

$\therefore \lim _{x \rightarrow 0} \frac{(\cos 3 x-\cos 5 x)}{x^{2}}=8$

 

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