Category: Limits

Question Number 141779 by iloveisrael last updated on 23/May/21 $$\underset{x\to 0}{\lim }\frac{x^2}{x+2-2\sqrt{1+x}}=?$$ Answered by Willson last updated on 23/May/21 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{x}+\mathrm{2}−\mathrm{2}\sqrt{\mathrm{1}+\mathrm{x}}}\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2x}}{\mathrm{1}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\mathrm{x}}}}…

Question Number 141699 by cesarL last updated on 22/May/21 $$If{lim}_{x\to 1}=\frac{\sqrt[k]{x}-1}{x-1}=L\ne 0Find{lim}_{x\to 0}\frac{\sqrt{x+1}-1}{\sqrt[k]{x+1}-1}$$ Answered by iloveisrael last updated on 22/May/21 $$Given\underset{x\to 1}{\lim }\frac{\sqrt[k]{x}-1}{x-1}=Lequivalent$$$${to}\:\underset{{t}\rightarrow\mathrm{1}}…

Question Number 141627 by cherokeesay last updated on 21/May/21 Answered by MJS_new last updated on 21/May/21 $$\frac{\sin 2x+\sin 4x}{1+\cos 2x+\cos 4x}=\frac{-2\sin x\cos x(1-{4\cos }^{2}x)}{1+{2\cos }^{2}x(1-{4\sin }^{2}x)}=$$$$\:\:\:\:\:\left[\mathrm{sin}\:{x}\:=\frac{\mathrm{tan}\:{x}}{\:\sqrt{\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \:{x}}}\wedge\mathrm{cos}\:{x}\:=\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \:{x}}}\right] \…

Question Number 75939 by ahmadshahhimat775@gmail.com last updated on 21/Dec/19 Commented by JDamian last updated on 21/Dec/19 $$c){(n!)}^{\frac{1}{n}}$$ Commented by mathmax by abdo last…

Question Number 75930 by Rio Michael last updated on 21/Dec/19 $$considerthefunction$$$$f\left(x\right)=\{\begin{array}{l}1+2x^2,ifxisrational\\ 1+x^4,ifxisirrational\end{array}$$$$Usethesandwich\left(pinchin\right)theoremto$$$$provethat\underset{x\to 0}{\lim }f\left(x\right)=1.$$ Terms of Service…