Category: Limits

Question Number 114259 by bobhans last updated on 18/Sep/20 $$\underset{x\to 0}{\lim }\frac{x\mathrm{arc}\sin \left(x^2\right)}{x\cos x-\sin x}?$$ Answered by bemath last updated on 18/Sep/20 $$by{L}^{\prime }Hopital$$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{arc}\:\mathrm{sin}\:\left({x}^{\mathrm{2}}…

Question Number 179657 by mathlove last updated on 31/Oct/22 $$provethat$$$$\underset{n\to \mathrm{\infty }}{\lim }(\sqrt{\frac{1}{2}\times \sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}}}}\times \sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}}}}\times \cdot \cdot \cdot \cdot \cdot \underset{nterm}{\underbrace{\sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2+}\dots ..+\frac{1}{2}\sqrt{\frac{1}{2}}}}}}})=\frac{2}{\pi }$$ Commented by mr W last updated on 01/Nov/22 $${the}\:{question}\:{should}\:{be}:…

Question Number 114088 by bemath last updated on 17/Sep/20 $$\underset{x\to \mathrm{\infty }}{\lim }\frac{2x\cot \left(\frac{2}{x}\right)-3\cot \left(\frac{2}{x}\right)}{5x^2-2x}$$ Answered by Olaf last updated on 17/Sep/20 $$=\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\mathrm{2}}{{x}}\right)^{\mathrm{2}} }{\left(\frac{\mathrm{2}}{{x}}\right)}\left[\frac{\mathrm{2}{x}−\mathrm{3}}{\mathrm{5}{x}^{\mathrm{2}} −\mathrm{2}{x}}\right]…

Question Number 114075 by bemath last updated on 17/Sep/20 Answered by john santu last updated on 17/Sep/20 $$setting\frac{1}{x}=b\wedge b\to 0$$$$\underset{b\to 0}{\lim }(\frac{1}{\sin b}-\frac{\cos b}{\sin b})=\underset{b\to 0}{\lim }\left(\frac{2\sin {}^2\left(\frac{1}{2}b\right)}{\sin b}\right)$$$$\:\:=\:\mathrm{0}…

Question Number 114020 by bemath last updated on 16/Sep/20 $$\underset{d\to \mathrm{\infty }}{\lim }\sqrt{6d}\cos \left(\frac{2}{\sqrt{d}}\right)\sin \left(\frac{5}{\sqrt{d}}\right)?$$$$$$ Answered by Olaf last updated on 17/Sep/20 $$\sqrt{\mathrm{6}{d}}\mathrm{cos}\left(\frac{\mathrm{2}}{\:\sqrt{{d}}}\right)\mathrm{sin}\left(\frac{\mathrm{5}}{\:\sqrt{{d}}}\right)\:\underset{\infty} {\sim}\:\sqrt{\mathrm{6}{d}}\left(\mathrm{1}−\frac{\mathrm{2}}{{d}}\right)\left(\frac{\mathrm{5}}{\:\sqrt{{d}}}\right) \…