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 $$\frac{{xy}}{{y}−{x}^{\mathrm{2}} }\:=\:\frac{{r}\mathrm{cos}\theta.{r}\mathrm{sin}\theta}{{r}\mathrm{sin}\theta−{r}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \theta} \\ $$$$=\:\frac{\frac{\mathrm{1}}{\mathrm{2}}{r}\mathrm{sin}\left(\mathrm{2}\theta\right)}{\mathrm{sin}\theta−{r}\mathrm{cos}^{\mathrm{2}} \theta} \\ $$$$\underset{{r}\rightarrow\mathrm{0}}…
Question Number 131388 by Eric002 last updated on 04/Feb/21 $$\underset{{x}\rightarrow\mathrm{4}} {{lim}}\frac{\sqrt{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{16}}+\mathrm{4}}{\mathrm{4}{x}^{\mathrm{2}} −\mathrm{16}{x}}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{16}}−\mathrm{4}} \\ $$ Answered by bemath last updated on 04/Feb/21 $$\:\underset{{x}\rightarrow\mathrm{4}} {\mathrm{lim}}\left(\frac{\sqrt{\mathrm{2}{x}^{\mathrm{2}}…
Question Number 131386 by bemath last updated on 04/Feb/21 $$\:\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\frac{{x}^{\mathrm{5}} \:\mathrm{cos}\:\left(\frac{\mathrm{1}}{\pi{x}^{\mathrm{2}} }\right)+{x}^{\mathrm{6}} \:\mathrm{sin}\:\left(\frac{\mathrm{1}}{\pi{x}}\right)+\:\mathrm{7}}{\mid{x}\mid^{\mathrm{5}} +\mathrm{6}\mid{x}\mid+\mathrm{7}}=? \\ $$ Answered by liberty last updated on 04/Feb/21 $$\:\underset{{x}\rightarrow−\infty}…