Category: Limits

Question Number 108228 by bemath last updated on 15/Aug/20 $$\frac{⋎\mathcal{B}e\mathcal{M}ath⋎}{⋔}$$$$\underset{n\to \mathrm{\infty }}{\lim }{\left(\frac{n+\ln a}{n}\right)}^{\frac{n}{b}}?$$ Answered by Dwaipayan Shikari last updated on 15/Aug/20 $$\underset{{n}\rightarrow\infty}…

Question Number 173729 by a.lgnaoui last updated on 17/Jul/22 $$Evaluate$$$${lim}_{x\to 0}\frac{x-\frac{1}{\sqrt{x}-x}+1}{x^2+\frac{\sqrt{x}}{\sqrt{x}-1}-1}$$ Answered by blackmamba last updated on 17/Jul/22 $$\:\:{L}\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{{x}−\frac{\mathrm{1}}{\:\sqrt{{x}}−{x}}\:−\mathrm{1}}{{x}^{\mathrm{2}}…

Question Number 42636 by bohemian last updated on 30/Aug/18 $$\underset{n\to \mathrm{\infty }}{\lim }\underset{r=1}{\sum ^n}\frac{r}{n^2+n+r}$$$$$$ Commented by bohemian last updated on 30/Aug/18…

Question Number 173498 by blackmamba last updated on 12/Jul/22 $$\underset{x\to 0}{\lim }\frac{8\cot \left(x\right)+9\sin \left(\frac{1}{x}\right)}{12\csc \left(x\right)-4\sin \left(\frac{1}{x}\right)}=?$$ Commented by blackmamba last updated on 13/Jul/22 $$=\underset{x\to 0}{\lim }\frac{8\cos x\left(\frac{x}{\sin x}\right)+9x\sin \left(\frac{1}{x}\right)}{\frac{12x}{\sin x}-4x\sin \left(\frac{1}{x}\right)}$$$$\:=\:\frac{\mathrm{8}+\mathrm{0}}{\mathrm{12}−\mathrm{0}}\:=\:\frac{\mathrm{8}}{\mathrm{12}}=\frac{\mathrm{2}}{\mathrm{3}} \…

Question Number 107930 by bemath last updated on 13/Aug/20 $$\frac{\mathbb{B}e\mathbb{M}ath}{\bullet \cap \bullet }$$$$\underset{x\to 0}{\lim }{\left(\sin x\right)}^{\frac{1}{\ln \sqrt{x}}}?$$ Answered by bemath last updated on 13/Aug/20 $$\Rightarrow{L}=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\mathrm{sin}\:\right)^{\frac{\mathrm{1}}{\mathrm{ln}\:\sqrt{{x}}}}…