Solve this:
Q. $\underset{\theta \to 0}{\lim }1-\cos \theta \frac{\cos 3\theta }{{\theta }^2}\cos 9\theta$

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{L}=\underset{\mathrm{x}\to 0}{\lim } 1-\frac{\mathrm{cos\theta cos}3\mathrm{\theta cos}9\mathrm{\theta }}{{\mathrm{\theta }}^2} \\ \mathrm{Now} \mathrm{we} \mathrm{know} \mathrm{that} \cos 0=1 \\ \mathrm{Hence} \\ \underset{\mathrm{x}\to 0}{\lim } 1-\frac{\mathrm{cos\theta cos}3\mathrm{\theta cos}9\mathrm{\theta }}{{\mathrm{\theta }}^2}=1-\frac{1}{0}\to 1-\infty \to -\infty \\ \mathrm{Hence} \\ \underset{\mathrm{x}\to 0}{\lim } 1-\frac{\mathrm{cos\theta cos}3\mathrm{\theta cos}9\mathrm{\theta }}{{\mathrm{\theta }}^2} \mathrm{approaches} -\infty \end{gathered}$$

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