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Question-174331




Question Number 174331 by cortano1 last updated on 30/Jul/22
Commented by kaivan.ahmadi last updated on 30/Jul/22
∼^(hop)  lim_(x→∞)  (((3e^(3x) )/(2+e^(3x) ))/((2e^(2x) )/(3+e^(2x) )))=lim_(x→∞) ((3e^(3x) (3+e^(2x) ))/(2e^(2x) (2+e^(3x) )))    =lim_(x→∞)  ((3e^x (3+e^(2x) ))/(2(2+e^(3x) ))) ∼^(hop)   lim_(x→∞) ((9e^x +9e^(3x) )/(6e^(3x) ))=(3/2)
$$\overset{{hop}} {\sim}\:{li}\underset{{x}\rightarrow\infty} {{m}}\:\frac{\frac{\mathrm{3}{e}^{\mathrm{3}{x}} }{\mathrm{2}+{e}^{\mathrm{3}{x}} }}{\frac{\mathrm{2}{e}^{\mathrm{2}{x}} }{\mathrm{3}+{e}^{\mathrm{2}{x}} }}={li}\underset{{x}\rightarrow\infty} {{m}}\frac{\mathrm{3}{e}^{\mathrm{3}{x}} \left(\mathrm{3}+{e}^{\mathrm{2}{x}} \right)}{\mathrm{2}{e}^{\mathrm{2}{x}} \left(\mathrm{2}+{e}^{\mathrm{3}{x}} \right)} \\ $$$$ \\ $$$$={li}\underset{{x}\rightarrow\infty} {{m}}\:\frac{\mathrm{3}{e}^{{x}} \left(\mathrm{3}+{e}^{\mathrm{2}{x}} \right)}{\mathrm{2}\left(\mathrm{2}+{e}^{\mathrm{3}{x}} \right)}\:\overset{{hop}} {\sim} \\ $$$${li}\underset{{x}\rightarrow\infty} {{m}}\frac{\mathrm{9}{e}^{{x}} +\mathrm{9}{e}^{\mathrm{3}{x}} }{\mathrm{6}{e}^{\mathrm{3}{x}} }=\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$ \\ $$

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