Category: Limits

Question Number 55905 by gunawan last updated on 06/Mar/19 $$\underset{n\to \mathrm{\infty }}{\lim }(x_{2n}+x_{2n+1})=315$$$$\underset{n\to \mathrm{\infty }}{\lim }(x_{2n}+x_{2n-1})=2016$$$$\underset{n\to \mathrm{\infty }}{\lim }\frac{x_{2n}}{x_{2n+1}}=\dots$$…

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Question Number 121222 by benjo_mathlover last updated on 06/Nov/20 $$\underset{x\to -\mathrm{\infty }}{\lim }\frac{\sqrt{1+{\mathrm{x}}^2}-\mathrm{x}+1}{\mathrm{x}+1}?$$ Answered by liberty last updated on 06/Nov/20 $$\:\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\frac{\mid\mathrm{x}\mid\sqrt{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }}−\mathrm{x}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{x}}\right)}{\mathrm{x}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{x}}\right)}\:= \…

Question Number 186748 by EnterUsername last updated on 09/Feb/23 $$\mathrm{Let}f:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}\mathrm{be}\mathrm{a}\mathrm{function}\mathrm{satisfying}\mathrm{the}\mathrm{relation}$$$$f(x.f\left(\mathrm{y}\right))=f\left(x\mathrm{y}\right)+x\mathrm{for}\mathrm{all}x,\mathrm{y}\in {\mathbb{R}}^{+}.\mathrm{Then}$$$$\underset{x\to 0}{\lim }\left(\frac{{\left(f\left(x\right)\right)}^{1/3}-1}{{\left(f\left(x\right)\right)}^{1/2}-1}\right)=$$$$\left(\mathrm{A}\right)1\left(\mathrm{B}\right)\frac{1}{2}$$$$\left(\mathrm{C}\right)\:\:\frac{\mathrm{2}}{\mathrm{3}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\:\frac{\mathrm{3}}{\mathrm{2}} \…