Question Number 184901 by mathlove last updated on 13/Jan/23 $$\underset{\frac{\mathrm{2}}{{x}}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\mathrm{100}+{x}}{\mathrm{101}{x}} \\ $$ Commented by SEKRET last updated on 13/Jan/23 $$\boldsymbol{\mathrm{x}}\rightarrow\mathrm{1} \\ $$ Answered by…
Question Number 119356 by Lordose last updated on 23/Oct/20 Commented by Dwaipayan Shikari last updated on 23/Oct/20 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{{x}}{log}\left(\frac{{x}^{{x}} }{{x}!}\right) \\ $$$$=\frac{\mathrm{1}}{{x}}.{log}\left(\frac{{x}^{{x}} {e}^{{x}} }{{x}^{{x}} .\sqrt{\mathrm{2}\pi{x}}}\right)\left({Stirling}'{s}\:{approximation}\:\underset{{n}\rightarrow\infty}…
Question Number 184891 by mathlove last updated on 13/Jan/23 $$\underset{{x}\rightarrow\pi} {\mathrm{lim}}\:\frac{{x}−\pi}{{sinx}}=? \\ $$ Answered by Mathspace last updated on 13/Jan/23 $${chang}.\:{x}−\pi={t}\:{give} \\ $$$${lim}_{{x}\rightarrow\pi} \frac{{x}−\pi}{{sinx}}={lim}_{{t}\rightarrow\mathrm{0}} \frac{{t}}{{sin}\left(\pi+{t}\right)}…
Question Number 184881 by mathlove last updated on 13/Jan/23 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\sqrt{\mathrm{2}}{sinx}}{{x}}\right)^{\frac{{sin}\frac{{x}}{\mathrm{2}}{cos}\frac{{x}}{\mathrm{2}}}{{x}}} =? \\ $$ Answered by aba last updated on 13/Jan/23 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\mathrm{cos}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)}{\mathrm{x}}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\left(\mathrm{x}\right)}{\mathrm{2x}}=\frac{\mathrm{1}}{\mathrm{2}} \\…
Question Number 184883 by mathlove last updated on 13/Jan/23 $$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\frac{\sqrt{{x}−\mathrm{2}}}{{x}−\mathrm{2}}=? \\ $$ Answered by aba last updated on 13/Jan/23 $$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\frac{\sqrt{\mathrm{x}−\mathrm{2}}}{\mathrm{x}−\mathrm{2}}=\underset{{x}\rightarrow\mathrm{2}^{+} } {\mathrm{lim}}\frac{\mathrm{1}}{\:\sqrt{\mathrm{x}−\mathrm{2}}}=\frac{\mathrm{1}}{\mathrm{0}^{+} }=+\infty…
Question Number 184880 by mathlove last updated on 13/Jan/23 $$\underset{{x}\rightarrow\frac{\mathrm{1}}{\pi}} {\mathrm{lim}}\:\:\frac{\pi^{−\mathrm{1}} −{x}}{\frac{\mathrm{1}−\left({x}\pi\right)^{\mathrm{2}} }{\pi^{\mathrm{2}} }}=? \\ $$ Answered by aba last updated on 13/Jan/23 $$\underset{{x}\rightarrow\frac{\mathrm{1}}{\pi}} {\mathrm{lim}}\frac{\pi^{−\mathrm{1}}…
Question Number 184882 by mathlove last updated on 13/Jan/23 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{ln}\left(\mathrm{7}{x}+\mathrm{1}\right)}{{x}}=? \\ $$ Answered by aba last updated on 13/Jan/23 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{ln}\left(\mathrm{7x}+\mathrm{1}\right)}{\mathrm{x}}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}7}\frac{\mathrm{ln}\left(\mathrm{7x}+\mathrm{1}\right)}{\mathrm{7x}}=\mathrm{7} \\ $$…
Question Number 184879 by mathlove last updated on 13/Jan/23 $$\underset{{x}\rightarrow\frac{\mathrm{1}}{\mathrm{4}}} {\mathrm{lim}}\:\mathrm{3}{x}=\frac{\mathrm{3}}{\mathrm{4}}\:\:\:\:\:\:\:\:\:\:\:{x}=? \\ $$ Answered by aba last updated on 13/Jan/23 $$\mathrm{x}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$ Terms of…
Question Number 184878 by mathlove last updated on 13/Jan/23 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{ln}\left(\mathrm{1}+{sinx}\right)}{{sinx}}=? \\ $$$${with}\:{out}\:{H}'{L}\:{Roule} \\ $$ Answered by aba last updated on 13/Jan/23 $$\mathrm{let}\:\mathrm{t}=\mathrm{sin}\left(\mathrm{x}\right) \\ $$$$\mathrm{x}\rightarrow\mathrm{0},\:\mathrm{t}\rightarrow\mathrm{0}…
Question Number 184797 by Spillover last updated on 11/Jan/23 $${Show}\:{that}\: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{e}^{\frac{\mathrm{1}}{{x}}} −\mathrm{1}}{{e}^{\frac{\mathrm{1}}{{x}}} +\mathrm{1}}\:\:\:{does}\:{not}\:{exist} \\ $$ Commented by MJS_new last updated on 11/Jan/23 $$\mathrm{let}\:{x}=\frac{\mathrm{1}}{{t}}…