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 117545 by Lordose last updated on 12/Oct/20

$$\underset{x\to 0^{+}}{\lim }{(1+{\tan }^2\left(\sqrt{\mathrm{x}}\right))}^{\frac{1}{2\mathrm{x}}}$$

Answered by TANMAY PANACEA last updated on 12/Oct/20

$$lny=\underset{x\to 0+}{\lim }\frac{ln(1+{tan}^2\left(\sqrt{x)}\right)}{{tan}^2\left(\sqrt{x}\right)}\times \frac{tan\left(\sqrt{x}\right)}{\sqrt{x}}\times \frac{tan\left(\sqrt{x}\right)}{\sqrt{x}}\times \frac{1}{2}$$$$lny=\frac{1}{2}\to y=e^{\frac{1}{2}}$$

Answered by Olaf last updated on 12/Oct/20

$$\frac{1}{2x}\ln (1+{\tan }^2\left(\sqrt{x}\right))\underset{0}{\sim }\frac{1}{2x}\ln (1+{\left(\sqrt{x}\right)}^2)\underset{0}{\sim }\frac{1}{2x}\times x=\frac{1}{2}$$$$\ln {(1+{\tan }^2\left(\sqrt{x}\right))}^{\frac{1}{2x}}\underset{0}{\sim }\frac{1}{2}$$$$\Rightarrow \underset{x\to 0^{+}}{\lim }{(1+{\tan }^2\left(\sqrt{x}\right))}^{\frac{1}{2x}}=e^{\frac{1}{2}}$$

Answered by Dwaipayan Shikari last updated on 12/Oct/20

$$\underset{x\to 0^{+}}{\lim }{(1+{tan}^2\sqrt{x})}^{\frac{1}{2x}}=\underset{x\to 0^{+}}{\lim }{(1+{\left(\sqrt{x}\right)}^2)}^{\frac{1}{2x}}={\left({(1+x)}^{\frac{1}{x}}\right)}^{\frac{1}{2}}=\sqrt{e}$$$$tan\sqrt{x}\to \sqrt{x}$$

Answered by bobhans last updated on 12/Oct/20

$$\mathrm{L}=\underset{x\to 0^{+}}{\lim }{(1+\tan {}^2\left(\sqrt{\mathrm{x}}\right))}^{\frac{1}{2\mathrm{x}}}$$$$\mathrm{L}={\mathrm{e}}^{\underset{x\to 0^{+}}{\lim }(1+\tan {}^2\left(\sqrt{\mathrm{x}}\right)-1)\times \frac{1}{2\mathrm{x}}}$$$$\mathrm{L}={\mathrm{e}}^{\underset{x\to 0^{+}}{\lim }\left(\frac{\tan {}^2\left(\sqrt{\mathrm{x}}\right)}{2{\left(\sqrt{\mathrm{x}}\right)}^2}\right)}={\mathrm{e}}^{\frac{1}{2}\times {\left(\underset{x\to 0^{+}}{\lim }\frac{\tan \sqrt{\mathrm{x}}}{\sqrt{\mathrm{x}}}\right)}^2}$$$$\mathrm{L}={\mathrm{e}}^{\frac{1}{2}}=\sqrt{\mathrm{e}}.$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com