If the solve the problem

Question:

If $\lim _{x \rightarrow 0}\left\{\frac{1}{x^{8}}\left(1-\cos \frac{x^{2}}{2}-\cos \frac{x^{2}}{4}+\cos \frac{x^{2}}{2} \cos \frac{x^{2}}{4}\right)\right\}=2^{-k}$

then the value of $\mathrm{k}$ is_______________

Solution:

$\lim _{x \rightarrow 0}\left\{\frac{1}{x^{8}}\left(1-\cos \frac{x^{2}}{2}-\cos \frac{x^{2}}{4}+\cos \frac{x^{2}}{2} \cos \frac{x^{2}}{4}\right)\right\}=2^{-k}$

$\Rightarrow \lim _{x \rightarrow 0} \frac{\left(1-\cos \frac{x^{2}}{2}\right)}{4\left(\frac{x^{2}}{2}\right)^{2}} \frac{\left(1-\cos \frac{x^{2}}{4}\right)}{16\left(\frac{x^{2}}{4}\right)^{2}}=\frac{1}{8} \times \frac{1}{32}=2^{-k}$

$\Rightarrow 2^{-8}=2^{-\mathrm{k}} \Rightarrow \mathrm{k}=8$

Leave a comment

Close

Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.

Download Now