$\lim _{x \rightarrow 4} \frac{\cos 7 x^{\circ}-\cos 2 x^{\circ}}{x^2}$ is

$\lim _{x \rightarrow 4} \frac{\cos 7 x^{\circ}-\cos 2 x^{\circ}}{x^2}$ is
  1. $\frac{-45}{2} \pi^2$
  2. $\frac{-45}{2} \pi$
  3. $\frac{-\pi^2}{1440}$
  4. $\frac{-\pi^2}{2880}$

Solution

$\begin{aligned} & \lim _{x \rightarrow 0} \frac{\cos 7 x^{\circ}-\cos 2 x^{\circ}}{x^2} \\ & =\lim _{x \rightarrow 0} \frac{\cos \left(\frac{7 \pi}{180}\right) x-\cos \left(\frac{2 \pi}{180}\right) x}{x^2} \\ & =\frac{\left(\frac{2 \pi}{180}\right)^2-\left(\frac{7 \pi}{180}\right)^2}{2} \\ & \quad \ldots\left[\because \lim _{x \rightarrow 0} \frac{\cos m x-\cos n x}{x^2}=\frac{n^2-m^2}{2}\right] \\ & =\frac{-\pi^2}{1440}\end{aligned}$

Asked in: MHT CET 2023 (12 May Shift 1)

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