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 176375 by cortano1 last updated on 17/Sep/22

$$\underset{x\to 0^{+}}{\lim }\frac{2\cos {}^2\left(\frac{1}{\mathrm{x}}\right)-\sin \left(\frac{1}{\mathrm{x}}\right)+3}{\mathrm{x}+\sqrt{\mathrm{x}}}=?$$

Commented by a.lgnaoui last updated on 18/Sep/22

$$posonsX=\frac{1}{x}x=\frac{1}{X}$$$$\left[\frac{5-\sin {}^{2}X-\sin X}{\sqrt{X}+1}\right]\sqrt{X}$$$$\frac{5\sqrt{X}}{\sqrt{X}+1}-\sqrt{X}\sin X\left(\frac{\sin X+1}{\sqrt{X}+1}\right)$$$$$$$$siutt=\sqrt{X}$$$$\frac{5t}{t+1}-t\sin \left(t^2\right)\left(\frac{\sin \left(t^2\right)+1}{t+1}\right)$$$$$$$$\frac{5t}{t+1}-\frac{t}{t+1}[\sin \left(t^2\right)(\sin \left(t^2\right)+1]$$$$x\to 0+t\to +[\mathrm{\infty }$$$${lim}_{x\to 0+}={lim}_{t\to +\mathrm{\infty }}\frac{5t}{t+1}-\frac{t}{t+1}[\sin \left(t^2\right)(\sin \left(t^2\right)+1]$$$$=5-{lim}_{X\to +\mathrm{\infty }}\left[\sin \left(X\right)(\sin \left(X\right)+1)\right]$$$$X=(2k+1)\frac{\pi }{2}\sin X\to 1$$$$donc{lim}_{x\to 0+}\frac{2\cos {}^2\left(\frac{1}{x}\right)-\sin \left(\frac{1}{x}\right)}{x+\sqrt{x}}=5-2=3$$$$$$

Commented by peter frank last updated on 19/Sep/22

$$\mathrm{font}\mathrm{size}\mathrm{is}\mathrm{too}\mathrm{large}.\mathrm{please}\mathrm{minimize}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com