Category: Limits

Question Number 120277 by bemath last updated on 30/Oct/20 $$\underset{x\to \mathrm{\infty }}{\lim }{x}^3\{\sqrt{x^2+\sqrt{x^4+1}}-x\sqrt{2}\}?$$ Commented by benjo_mathlover last updated on 30/Oct/20 $$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}^{\mathrm{4}}…

Question Number 185776 by TUN last updated on 27/Jan/23 Answered by MJS_new last updated on 27/Jan/23 $$n!\sim \frac{n^n}{{\mathrm{e}}^n}\sqrt{2\pi n}$$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{\left(\mathrm{2}{n}\right)!}{{n}!{n}^{{n}} }\right)^{\mathrm{1}/{n}} \:=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{4}×\mathrm{2}^{\mathrm{1}/\left(\mathrm{2}{n}\right)}…

Question Number 54688 by pooja24 last updated on 09/Feb/19 $$f\left(x\right)=\frac{2\left[x\right]}{3x-\left[x\right]}examineitscontinuityatx=\frac{-1}{2}$$$$where\left[x\right]isgreatestintegerfunction$$ Commented by maxmathsup by imad last updated on 10/Feb/19 $${we}\:{have}\:{f}\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)\:=\frac{\mathrm{2}\left(−\mathrm{1}\right)}{−\frac{\mathrm{3}}{\mathrm{2}}−\left(−\mathrm{1}\right)}\:=\frac{−\mathrm{2}}{−\frac{\mathrm{3}}{\mathrm{2}}+\mathrm{1}}\:=\frac{−\mathrm{2}}{−\frac{\mathrm{1}}{\mathrm{2}}}\:=\mathrm{4} \…

Question Number 120201 by bemath last updated on 30/Oct/20 $$\underset{x\to \mathrm{\infty }}{\lim }\frac{\ln (x+e^x+e^{2x})}{x}=?$$ Answered by benjo_mathlover last updated on 30/Oct/20 $$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{ln}\:\left({x}+{e}^{{x}} +{e}^{\mathrm{2}{x}}…

Question Number 120067 by bramlexs22 last updated on 29/Oct/20 $$\left(i\right)\underset{x\to 0}{\lim }\frac{x^2\sin \left(1/x\right)}{\tan x}$$$$\left(ii\right)Without{L}^{\prime }Hopitalrule$$$$\underset{x\to 0^{+}}{\lim }\frac{1-\cos x-x\sin x}{2-2\cos x-\sin {}^{2}x}$$ Answered by Olaf last…