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Question Number 29689 by mrW2 last updated on 11/Feb/18

$$find\underset{x\to 0}{\lim }{\left(\frac{2}{1+a^x}\right)}^{\frac{1}{x}}=?$$

Commented by abdo imad last updated on 13/Feb/18

$$letputA\left(x\right)={\left(\frac{2}{1+a^x}\right)}^{\frac{1}{x}}\Rightarrow ln\left(A\left(x\right)\right)=\frac{1}{x}ln\left(\frac{2}{1+a^x}\right)$$$$=\frac{1}{x}(ln\left(2\right)-ln(1+a^x))=\frac{1}{x}(ln\left(2\right)-ln(2+a^x-1))$$$$=\frac{1}{x}(-ln(1+\frac{a^x-1}{2}))\Rightarrow A\left(x\right)=e^{\frac{-1}{x}ln(1+\frac{a^x-1}{2})}forx\in v\left(0\right)$$$$\frac{a^x-1}{2}\in v\left(0\right)andln(1+u)\sim u\Rightarrow ln(1+\frac{a^x-1}{2})\sim \frac{a^x-1}{2}\Rightarrow$$$$-\frac{1}{x}ln(...)\sim \frac{1-a^x}{2x}byhospitaltheorem$$$${lim}_{x\to 0}\frac{1-a^x}{2x}={lim}_{x\to 0}\frac{1-e^{xlna}}{2x}={lim}_{x\to 0}\frac{-lna{e}^{xlna}}{2}$$$$=ln\left(\frac{1}{\sqrt{a}}\right)\Rightarrow {lim}_{x\to 0}A\left(x\right)=\frac{1}{\sqrt{a}}.$$

Commented by mrW2 last updated on 13/Feb/18

$$thankssir!$$

Answered by ajfour last updated on 11/Feb/18

$$let\underset{x\to 0}{\lim }{\left(\frac{2}{1+a^x}\right)}^{1/x}=L$$$$\ln L=\underset{x\to 0}{\lim }\left[\frac{\ln (1+\frac{1-a^x}{1+a^x})}{x}\right]$$$$=\underset{x\to 0}{\lim }\left[\frac{\left(\frac{1-a^x}{1+a^x}\right)}{x}\right]$$$$=\underset{x\to 0}{\lim }\frac{(-x\ln a-\frac{x^2}{2}{\left(\ln a\right)}^2-...)}{x(1+a^x)}$$$$=\underset{x\to 0}{\lim }[-\ln a\left(\frac{1+\frac{x\ln a}{2}+..}{1+a^x}\right)]$$$$\Rightarrow \ln L=-\frac{\ln a}{2}$$$$\Rightarrow \mathit{L}=\frac{1}{\sqrt{\mathit{a}}}.$$

Commented by 803jaideep@gmail.com last updated on 11/Feb/18

$$\mathrm{are}\mathrm{you}\mathrm{guys}\mathrm{teacher}?....\mathrm{you}\mathrm{solve}\mathrm{all}\mathrm{ques}\mathrm{easily}.....\mathrm{or}\mathrm{a}\mathrm{dtudent}\mathrm{of}+1+2?$$

Commented by ajfour last updated on 11/Feb/18

$$iteachphysicsfor10+1,10+2.whataboutyousir?$$

Commented by 803jaideep@gmail.com last updated on 11/Feb/18

$$\mathrm{im}\mathrm{a}+1\mathrm{student}$$

Commented by math solver last updated on 11/Feb/18

$$\mathrm{wow}!\mathrm{still}\mathrm{solving}\mathrm{maths}\mathrm{q}.\mathrm{efficiently}.$$$$\mathrm{my}\mathrm{phy}.\mathrm{teacher}\mathrm{should}\mathrm{learn}\mathrm{then}!$$

Commented by math solver last updated on 11/Feb/18

$$\mathrm{by}\mathrm{reading}\mathrm{the}\mathrm{answers}\mathrm{of}\mathrm{mrw}2\mathrm{sir}$$$$\mathrm{i}\mathrm{assumed}\mathrm{he}\mathrm{is}\mathrm{a}\mathrm{professor}\mathrm{at}\mathrm{some}$$$$\mathrm{university}\mathrm{but}\mathrm{later}\mathrm{i}\mathrm{came}\mathrm{to}\mathrm{know}$$$$\mathrm{he}\mathrm{is}\mathrm{an}\mathrm{engineer}\mathrm{and}\mathrm{have}\mathrm{very}\mathrm{little}$$$$\left(\mathrm{almost}\mathrm{no}\right)\mathrm{use}\mathrm{of}\mathrm{phy}.,\mathrm{maths}.$$

Commented by 803jaideep@gmail.com last updated on 11/Feb/18

$$\mathrm{are}\mathrm{u}\mathrm{also}\mathrm{teacher}?$$

Commented by math solver last updated on 11/Feb/18

$$\mathrm{no},\mathrm{i}\mathrm{am}\mathrm{also}+1\mathrm{student}.$$$$\mathrm{moving}\mathrm{to}+2\mathrm{this}\mathrm{april}.$$

Commented by math solver last updated on 11/Feb/18

$$\mathrm{just}\mathrm{go}\mathrm{to}\mathrm{question}\mathrm{ID}22537.$$

Commented by mrW2 last updated on 11/Feb/18

Thank you ajfour sir! I guess the answer should be 1/√a, see Q29633, but I had no proof. Now it is sure.

Commented by 803jaideep@gmail.com last updated on 11/Feb/18

$$\mathrm{is}\mathrm{ur}\mathrm{target}\mathrm{iit}?...\mathrm{u}\mathrm{f}\mathrm{vry}\mathrm{intelligent}\mathrm{at}16\mathrm{im}\mathrm{sldo}16$$

Commented by math solver last updated on 11/Feb/18

$$\mathrm{yes}!$$$$\mathrm{btw},\mathrm{i}\mathrm{am}\mathrm{sorry}\mathrm{i}\mathrm{am}\mathrm{a}\mathrm{nerd}\mathrm{infront}\mathrm{of}$$$$\mathrm{these}\mathrm{guys}\mathrm{asking}\mathrm{silly}\mathrm{questions}:($$

Commented by 803jaideep@gmail.com last updated on 11/Feb/18

$$\mathrm{thn}\mathrm{think}\mathrm{about}\mathrm{me}...\mathrm{i}\mathrm{also}\mathrm{target}\mathrm{iit}$$

Commented by 803jaideep@gmail.com last updated on 11/Feb/18

$$\mathrm{fir}\mathrm{bhi}?...\mathrm{avg}?...\mathrm{me}\mathrm{abt}6\mathrm{hrs}\mathrm{generally}$$

Commented by 803jaideep@gmail.com last updated on 11/Feb/18

$$\mathrm{im}\mathrm{in}+1\mathrm{moving}+2\mathrm{this}\mathrm{april}$$

Commented by 803jaideep@gmail.com last updated on 11/Feb/18

$$\mathrm{how}\mathrm{many}\mathrm{hours}\mathrm{a}\mathrm{day}\mathrm{u}\mathrm{study}?$$

Commented by math solver last updated on 11/Feb/18

$$\mathrm{its}\mathrm{not}\mathrm{fixed},\mathrm{even}\mathrm{sometimes}\mathrm{it}\mathrm{goes}$$$$\mathrm{like}\mathrm{i}{}^{\prime }\mathrm{nt}\mathrm{studied}\mathrm{for}\mathrm{a}\mathrm{whole}\mathrm{day}.$$

Commented by 803jaideep@gmail.com last updated on 11/Feb/18

$$\mathrm{im}\mathrm{general}\mathrm{category}...\mathrm{im}\mathrm{from}$$$$\mathrm{punjab}\mathrm{patiala}\mathrm{by}\mathrm{d}\mathrm{way}$$

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