Question Number 78635 by Pratah last updated on 19/Jan/20 Commented by john santu last updated on 19/Jan/20 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{\underset{{n}} {\overset{\mathrm{2}{n}} {\int}}\:\left(\frac{{x}}{{x}^{\mathrm{5}} +\mathrm{1}}\right){dx}\:}{{n}^{−\mathrm{3}} }\right)=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left(\frac{\mathrm{2}{n}}{\mathrm{32}{n}^{\mathrm{5}} +\mathrm{1}}\right).\mathrm{2}−\left(\frac{{n}}{{n}^{\mathrm{5}}…
Question Number 78596 by jagoll last updated on 19/Jan/20 $$ \\ $$$$ \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{1}−\mathrm{3tan}\:^{\mathrm{2}} \mathrm{x}\right)^{\frac{\mathrm{2}}{\mathrm{sin}\:^{\mathrm{2}} \:\mathrm{3x}}\:\:\:} =\:?\: \\ $$ Commented by mathmax by abdo…
Question Number 413 by 123456 last updated on 25/Jan/15 $$\underset{\left({x},{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)} {\mathrm{lim}}\:\:\:\frac{\left({x}+{y}\right)\left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right)}{\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }} \\ $$ Answered by prakash jain last updated on 31/Dec/14…
Question Number 412 by 123456 last updated on 25/Jan/15 $$\underset{\left({x},{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)} {\mathrm{lim}}\:\:\frac{\mathrm{sin}\:\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)}{\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }} \\ $$ Answered by prakash jain last updated on 31/Dec/14…
Question Number 394 by prakash jain last updated on 27/Dec/14 $${f}\left({x}\right)=\underset{{i}=\mathrm{1}} {\overset{\infty} {\sum}}{a}_{{i}} {x}^{{i}} \\ $$$${a}_{{i}} \:\:\mathrm{A}\:\mathrm{constant}\:\mathrm{independent}\:\mathrm{of}\:{x} \\ $$$$\mathrm{I}{s}\:\mathrm{the}\:\mathrm{below}\:\mathrm{statement}\:\mathrm{correct}? \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{f}\left({x}\right)=\mathrm{0} \\ $$$$\mathrm{If}\:\mathrm{not}\:\mathrm{give}\:\mathrm{an}\:\mathrm{example}. \\…
Question Number 379 by userid1 last updated on 25/Jan/15 $$\mathrm{Find}\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{{e}^{\mathrm{1}/{x}} −\mathrm{1}}{{e}^{\mathrm{1}/{x}} +\mathrm{1}}\right) \\ $$ Commented by 123456 last updated on 25/Dec/14 $$\underset{{x}\rightarrow\mathrm{0}+} {\mathrm{lim}}\frac{{e}^{\frac{\mathrm{1}}{{x}}} −\mathrm{1}}{{e}^{\frac{\mathrm{1}}{{x}}}…
Question Number 131444 by rs4089 last updated on 04/Feb/21 Answered by Olaf last updated on 05/Feb/21 $$\mathrm{polar}\:\mathrm{coordinates}\:: \\ $$$$\frac{{xy}+\mathrm{2}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }\:=\:\frac{{r}\mathrm{cos}\theta.{r}\mathrm{sin}\theta+\mathrm{2}}{{r}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \theta+{r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \theta}…
Question Number 131445 by rs4089 last updated on 04/Feb/21 Answered by mathmax by abdo last updated on 04/Feb/21 $$\mid\left(\mathrm{x}+\mathrm{y}\right)\mathrm{sin}\left(\frac{\mathrm{1}}{\mathrm{x}+\mathrm{y}}\right)\mid\leqslant\mid\mathrm{x}+\mathrm{y}\mid\:\Rightarrow\mathrm{lim}_{\left(\mathrm{x},\mathrm{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)} \:\:\left(\mathrm{x}+\mathrm{y}\right)\mathrm{sin}\left(\frac{\mathrm{1}}{\mathrm{x}+\mathrm{y}}\right)=\mathrm{0} \\ $$ Terms of Service…
Question Number 370 by 123456 last updated on 25/Jan/15 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left[\frac{{f}\left({a}+\frac{\mathrm{1}}{{n}}\right)}{{f}\left({a}−\frac{\mathrm{1}}{{n}}\right)}\right]^{{n}} \\ $$ Answered by prakash jain last updated on 24/Dec/14 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}ln}\:{y}=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{ln}\:{f}\left({a}+\frac{\mathrm{1}}{{n}}\right)−\mathrm{ln}\:{f}\left({a}−\frac{\mathrm{1}}{{n}}\right)}{\mathrm{1}/{n}} \\…
Question Number 131443 by rs4089 last updated on 04/Feb/21 Answered by Olaf last updated on 05/Feb/21 $$\frac{{xy}}{{y}−{x}^{\mathrm{2}} }\:=\:\frac{{r}\mathrm{cos}\theta.{r}\mathrm{sin}\theta}{{r}\mathrm{sin}\theta−{r}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \theta} \\ $$$$=\:\frac{\frac{\mathrm{1}}{\mathrm{2}}{r}\mathrm{sin}\left(\mathrm{2}\theta\right)}{\mathrm{sin}\theta−{r}\mathrm{cos}^{\mathrm{2}} \theta} \\ $$$$\underset{{r}\rightarrow\mathrm{0}}…