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 $$\:\underset{{x}\rightarrow−\infty}…
Question Number 131383 by EDWIN88 last updated on 04/Feb/21 $$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\frac{{x}^{\mathrm{2}} }{{x}−\mathrm{1}}\right)^{\mathrm{tan}\:\left(\frac{\mathrm{1}}{\:\sqrt{{x}}}\right)} =? \\ $$ Answered by liberty last updated on 04/Feb/21 $$\mathrm{L}=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{x}−\mathrm{1}}\right)^{\mathrm{tan}\:\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{x}}}\right)}…
Question Number 131373 by EDWIN88 last updated on 10/Feb/21 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left[\:\sqrt[{\mathrm{3}}]{{x}^{\mathrm{3}} +\mathrm{3}{x}^{\mathrm{2}} }\:+\sqrt{{x}^{\mathrm{2}} −\mathrm{2}{x}}\:−\mathrm{2}{x}\:\right]=? \\ $$ Answered by aleks041103 last updated on 04/Feb/21 $${L}=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}\left[\:\sqrt[{\mathrm{3}}]{{x}^{\mathrm{3}}…
Question Number 131369 by liberty last updated on 04/Feb/21 $$\mathrm{If}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\left(\mathrm{1}−\mathrm{cos}\:\mathrm{x}\right)}{\mathrm{x}^{\mathrm{4}} }\:=\:\frac{\mathrm{m}}{\mathrm{n}}\:\mathrm{where}\:\mathrm{m}\:\mathrm{and}\:\mathrm{n} \\ $$$$\mathrm{are}\:\mathrm{relative}\:\mathrm{prime}\:\mathrm{positive}\:\mathrm{integer}\: \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\left(\mathrm{m}^{\mathrm{2}} +\mathrm{n}^{\mathrm{2}} \right)\:\mathrm{equals} \\ $$ Answered by EDWIN88 last updated…
Question Number 131371 by EDWIN88 last updated on 04/Feb/21 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}\left(\mathrm{tan}\:{x}−\mathrm{sin}\:{x}\right)−{x}^{\mathrm{3}} }{{x}^{\mathrm{5}} }\:=? \\ $$ Answered by liberty last updated on 04/Feb/21 $$\mathrm{let}\:\mathrm{x}\:=\:\mathrm{2t}\: \\ $$$$\mathrm{L}=\underset{\mathrm{t}\rightarrow\mathrm{0}}…