Category: Limits

Question Number 197281 by cortano12 last updated on 12/Sep/23 $$\underset{x\to 0}{\lim }\frac{\sin \mathrm{x}-\mathrm{x}+{2\mathrm{x}}^5}{{3\mathrm{x}}^3}=?$$ Answered by MM42 last updated on 12/Sep/23 $${lim}_{{x}\rightarrow\mathrm{0}} \:\frac{−\frac{\mathrm{1}}{\mathrm{6}}{x}^{\mathrm{3}} +\mathrm{2}{x}^{\mathrm{5}}…

Question Number 197110 by cortano12 last updated on 08/Sep/23 Answered by MathematicalUser2357 last updated on 10/Sep/23 $$\mathrm{let}f\left(x\right)=\frac{\sqrt[3]{{(x^2+1)}^2}-2\sqrt[3]{x^2+3}+\sqrt[3]{4}}{{(x-1)}^2}$$$$\mathrm{Then}\:\underset{{x}\rightarrow\mathrm{1}^{+} } {\mathrm{lim}}{f}\left({x}\right)=\infty…

Question Number 197008 by tri26112004 last updated on 06/Sep/23 Answered by AST last updated on 06/Sep/23 $$\frac{{x}^{\mathrm{2}} −{y}^{\mathrm{2}} ={x}^{\mathrm{2}} +{y}^{\mathrm{2}} −\mathrm{2}{y}^{\mathrm{2}} }{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }=\mathrm{1}−\frac{\mathrm{2}{y}^{\mathrm{2}} }{{x}^{\mathrm{2}}…

Question Number 196959 by cortano12 last updated on 05/Sep/23 $$\underset{―}{}$$ Answered by horsebrand11 last updated on 05/Sep/23 $$\:\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{sin}\:\mathrm{2x}−\mathrm{2x}}{\mathrm{x}^{\mathrm{3}} }\:\right)+\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{ax}+\mathrm{2x}}{\mathrm{x}^{\mathrm{3}} }\right)=\mathrm{1}−\mathrm{b}…

Question Number 196534 by cortano12 last updated on 27/Aug/23 $$\underset{x\to \mathrm{\infty }}{\lim }\mathrm{x}\sqrt{1-\cos \left(\frac{\pi }{\mathrm{x}}\right)}=?$$ Answered by Mathspace last updated on 27/Aug/23 $$1-cosu\sim \frac{u^2}{2}(u\to o)\Rightarrow$$$$\mathrm{1}−{cos}\left(\frac{\pi}{{x}}\right)\sim\frac{\pi^{\mathrm{2}} }{\mathrm{2}{x}^{\mathrm{2}}…