$\lim _{x \rightarrow 0} \frac{\cos (m x)-\cos (n x)}{x^2}=$

$\lim _{x \rightarrow 0} \frac{\cos (m x)-\cos (n x)}{x^2}=$
  1. $\frac{\mathrm{m}^2-\mathrm{n}^2}{2}$
  2. $m^2-n^2$
  3. $\frac{\mathrm{n}^2-\mathrm{m}^2}{2}$
  4. $n^2-m^2$

Solution

$\begin{aligned} & \lim _{x \rightarrow 0} \frac{\cos (m x)-\cos (n x)}{x^2} \\ & =\lim _{x \rightarrow 0} \frac{\left[-2 \sin \frac{(m+n) x}{2} \sin \frac{(m-n) x}{2}\right]}{x^2} \\ & =-2 \lim _{x \rightarrow 0}\left[\frac{\sin \left(\frac{m+n}{2}\right) x}{\left(\frac{m+n}{2}\right) x} \times\left(\frac{m+n}{2}\right)\right]\left[\frac{\sin \left(\frac{m-n}{2}\right) x}{\left(\frac{m-n}{2}\right) x} \times\left(\frac{m-n}{2}\right)\right] \\ & =(-2)\left(\frac{m+n}{2}\right)\left(\frac{m-n}{2}\right)=\frac{m^2-n^2}{-2}=\frac{n^2-m^2}{2}\end{aligned}$

Asked in: MHT CET 2021 (22 Sep Shift 1)

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