Category: Limits

Question Number 123398 by bemath last updated on 25/Nov/20 $$\underset{x\to 0}{\lim }\frac{\sqrt[3]{1+3x}-\sqrt{1+2x}}{x^2}?$$ Answered by Dwaipayan Shikari last updated on 25/Nov/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}+\frac{\mathrm{3}{x}}{\mathrm{3}}+\frac{\mathrm{9}{x}^{\mathrm{2}} }{\mathrm{2}!}.\left(−\frac{\mathrm{2}}{\mathrm{9}}\right)−\mathrm{1}−\frac{\mathrm{2}{x}}{\mathrm{2}}+\frac{\mathrm{4}{x}^{\mathrm{2}}…

Question Number 57850 by Mikael_Marshall last updated on 13/Apr/19 $$\underset{n\to \mathrm{\infty }}{lim}\frac{8^n}{2^{n+1}+3^{n+2}}$$ Answered by tanmay.chaudhury50@gmail.com last updated on 13/Apr/19 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{8}^{{n}}…

Question Number 123349 by bemath last updated on 25/Nov/20 $$\underset{x\to \mathrm{\infty }}{\lim }[x\sqrt{\frac{x-1}{9x+2}}-\frac{x}{3}]?$$ Answered by liberty last updated on 25/Nov/20 $$\underset{x\to \mathrm{\infty }}{\lim }x[\sqrt{\frac{x-1}{9x+2}}-\frac{1}{3}]=$$$$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{{x}\:\left(\mathrm{3}\sqrt{{x}−\mathrm{1}}\:−\sqrt{\mathrm{9}{x}+\mathrm{2}}\right)}{\:\mathrm{3}\sqrt{\mathrm{9}{x}+\mathrm{2}}}\:=…

Question Number 188826 by mathlove last updated on 08/Mar/23 $$provethat$$$$\underset{n\to \mathrm{\infty }}{\lim }\frac{\sqrt[n(n+1)]{1!2!3!\cdot \cdot \cdot \cdot \cdot n!}}{\sqrt{n}}=e^{\frac{-3}{4}}$$ Commented by Frix last updated on 07/Mar/23 $$\mathrm{There}\mathrm{is}\mathrm{no}x\mathrm{and}\mathrm{why}\to 0?$$…

Question Number 188798 by mathlove last updated on 07/Mar/23 Answered by ARUNG_Brandon_MBU last updated on 07/Mar/23 $$\mathcal{L}=\underset{x\to 0}{\lim }\frac{2^x{3}^x{4}^x{5}^x\dots n^x-1}{x}$$$$\:\:\:\:\:=\underset{{x}\rightarrow\mathrm{0}}…