Category: Limits

Question Number 73561 by Rio Michael last updated on 13/Nov/19 $$findthevalueof\lambda forwhich$$$$f:x\to \{\begin{array}{l}2\lambda -x,ifx<1\\ {\lambda }^2+x-1,ifx>1\end{array}$$$$hasalimitasx\to 1$$ Answered by ajfour last updated on 13/Nov/19…

Question Number 73552 by Rio Michael last updated on 13/Nov/19 $$evaluate$$$$\underset{x\to +\mathrm{\infty }}{Lim}xln\left(\frac{x+1}{x}\right)$$ Commented by mathmax by abdo last updated on 13/Nov/19…

Question Number 73518 by aliesam last updated on 13/Nov/19 Commented by mathmax by abdo last updated on 13/Nov/19 $$letA\left(x\right)={\left(\frac{x+1}{x-1}\right)}^{2x-1}\Rightarrow A\left(x\right)=e^{(2x-1)ln\left(\frac{x+1}{x-1}\right)}$$$$={e}^{\left(\mathrm{2}{x}−\mathrm{1}\right){ln}\left(\frac{{x}−\mathrm{1}+\mathrm{2}}{{x}−\mathrm{1}}\right)} \:={e}^{\left(\mathrm{2}{x}−\mathrm{1}\right){ln}\left(\mathrm{1}+\frac{\mathrm{2}}{{x}−\mathrm{1}}\right)} \:\:{we}\:{have}\:{ln}\left(\mathrm{1}+\frac{\mathrm{2}}{{x}−\mathrm{1}}\right)\sim\frac{\mathrm{2}}{{x}−\mathrm{1}}\left({x}\rightarrow+\infty\right) \…

Question Number 138996 by bramlexs22 last updated on 21/Apr/21 $$\underset{x\to \pi /2}{\lim }\frac{\sqrt{2}-\sqrt{1+\sin \mathrm{x}}}{\sqrt{2}\cos {}^2\mathrm{x}}=?$$$$$$ Answered by EDWIN88 last updated on 21/Apr/21 $$\:\underset{{x}\rightarrow\pi/\mathrm{2}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{sin}\:\mathrm{x}}{\:\sqrt{\mathrm{2}}\left(\mathrm{1}−\mathrm{sin}\:\mathrm{x}\right)}.\underset{{x}\rightarrow\pi/\mathrm{2}}…

Question Number 138959 by mathlove last updated on 20/Apr/21 Answered by mathmax by abdo last updated on 20/Apr/21 $$\mathrm{a}^{\mathrm{x}^{\mathrm{2}} −\mathrm{1}} \:=\mathrm{e}^{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)\mathrm{loga}} \:\sim\mathrm{1}+\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)\mathrm{loga}\:\mathrm{and} \…