Question and Answers Forum

All Questions      Topic List

Limits Questions

Previous in All Question      Next in All Question      

Previous in Limits      Next in Limits      

Question Number 129731 by Adel last updated on 18/Jan/21

$$\underset{x\to \mathrm{\infty }}{\lim cos}\frac{\mathrm{x}}{2}\times \cos \frac{\mathrm{x}}{4}.......\cos \frac{\mathrm{x}}{2^{\mathrm{n}}}=?$$

Answered by Dwaipayan Shikari last updated on 18/Jan/21

$$cos\frac{x}{2}=\frac{sinx}{2sin\frac{x}{2}}$$$$\underset{n=1}{\prod ^{\mathrm{\infty }}}cos\left(\frac{x}{2^n}\right)=\underset{n\to \mathrm{\infty }}{\lim }\frac{sin\left(x\right)}{2sin\left(\frac{x}{2}\right)}.\frac{sin\left(\frac{x}{2}\right)}{2sin\left(\frac{x}{4}\right)}.\frac{sin\left(\frac{x}{4}\right)}{2sin\left(\frac{x}{8}\right)}...\frac{sin\left(\frac{x}{2^{n-1}}\right)}{2sin\left(\frac{x}{2^n}\right)}$$$$=\frac{sin\left(x\right)}{2^{n}sin\left(\frac{x}{2^n}\right)}=\frac{sinx}{2^n.\frac{x}{2^n}}=\frac{sinx}{x}\underset{n\to \mathrm{\infty }}{\lim }sin\left(\frac{x}{2^n}\right)\sim \frac{x}{2^n}$$

Commented by Adel last updated on 18/Jan/21

$$\mathrm{tanks}\mathrm{bro}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com