Category: Limits

Question Number 128604 by john_santu last updated on 08/Jan/21 $$\underset{x\to 1}{\lim }\frac{{\mathrm{x}}^{\mathrm{x}}-\mathrm{x}}{\ln \mathrm{x}-\mathrm{x}+1}?$$ Answered by liberty last updated on 09/Jan/21 $$\:\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{x}} \right)=\mathrm{x}^{\mathrm{x}} \left(\mathrm{ln}\:\mathrm{x}+\mathrm{1}\right) \…

Question Number 63054 by jimful last updated on 28/Jun/19 $$if\mathrm{\Sigma }\mid a_n\mid isconvergent,then$$$$provethatthereexists$$$$asubsequence\left\{n_k{a}_{n_k}\right\}with$$$$\underset{{k}\rightarrow\infty} {\mathrm{lim}}{n}_{{k}} {a}_{{n}_{{k}} } =\mathrm{0} \…

Question Number 62987 by aliesam last updated on 27/Jun/19 Terms of Service Privacy Policy Contact: info@tinkutara.com

Question Number 128439 by Study last updated on 07/Jan/21 $$li\underset{x\to \frac{\pi }{2}}{m}\frac{cosx}{1-sinx}=???$$ Answered by benjo_mathlover last updated on 07/Jan/21 $$\mathrm{let}\mathrm{x}=\frac{\pi }{2}+\mathrm{z}\wedge \mathrm{z}\to 0$$$$\:\underset{\mathrm{z}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}}+\mathrm{z}\right)}{\mathrm{1}−\mathrm{sin}\:\left(\mathrm{z}+\frac{\pi}{\mathrm{2}}\right)}=\underset{\mathrm{z}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{−\mathrm{sin}\:\mathrm{z}}{\mathrm{1}−\mathrm{cos}\:\mathrm{z}}…

Question Number 62747 by Mikael last updated on 24/Jun/19 $$Are\mathit{f},\mathit{g}:\mathbb{R}\to \mathbb{R}definedby$$$$f\left(x\right)=\{\begin{array}{l}0,x\in \mathbb{R}\mathrm{\setminus }\mathbb{Q}\\ x,x\in \mathbb{Q}\end{array}$$$$g\left(x\right)=\{\begin{array}{l}1,x=0\\ 0,x\ne 0\end{array}$$$$showthat\underset{\mathbf{x}\to 0}{\mathbf{lim}}\mathbf{f}\left(\mathbf{x}\right)=0and\underset{\mathbf{y}\to 0}{\mathbf{lim}}\mathbf{g}\left(\mathbf{y}\right)=0$$$$however\underset{\mathbf{x}\to 0}{\mathbf{lim}}\mathbf{g}\left(\mathbf{f}\left(\mathbf{x}\right)\right)doesnotexist.$$ Answered by…