Question and Answers Forum

All Questions      Topic List

Limits Questions

Previous in All Question      Next in All Question      

Previous in Limits      Next in Limits      

Question Number 112900 by bemath last updated on 10/Sep/20

 lim_(x→∞) (((x^3 −x^2 +1)/(2x^3 +x^2 −2)))^x ?

$$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{x}^{\mathrm{3}} −\mathrm{x}^{\mathrm{2}} +\mathrm{1}}{\mathrm{2x}^{\mathrm{3}} +\mathrm{x}^{\mathrm{2}} −\mathrm{2}}\right)^{\mathrm{x}} ? \\ $$

Commented by kaivan.ahmadi last updated on 10/Sep/20

∼lim_(x→∞) ((x^3 /(2x^3 )))^x   =((1/2))^(+∞) =0

$$\sim{lim}_{{x}\rightarrow\infty} \left(\frac{{x}^{\mathrm{3}} }{\mathrm{2}{x}^{\mathrm{3}} }\right)^{{x}} \:\:=\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{+\infty} =\mathrm{0} \\ $$

Answered by mathmax by abdo last updated on 10/Sep/20

let f(x) =(((x^3 −x^2  +1)/(2x^3  +x^2 −2)))^x  ⇒f(x) ={((1−(1/x)+(1/x^2 ))/(2+(1/x)−(2/x^3 )))}^x   ⇒f(x) =e^(xln(((1−(1/x)+(1/x^2 ))/(2+(1/x)−(2/x^3 )))))  ∼ e^(−xln2)  ⇒lim_(x→+∞) f(x) =0 and  lim_(x→−∞) f(x) =+∞

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)\:=\left(\frac{\mathrm{x}^{\mathrm{3}} −\mathrm{x}^{\mathrm{2}} \:+\mathrm{1}}{\mathrm{2x}^{\mathrm{3}} \:+\mathrm{x}^{\mathrm{2}} −\mathrm{2}}\right)^{\mathrm{x}} \:\Rightarrow\mathrm{f}\left(\mathrm{x}\right)\:=\left\{\frac{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }}{\mathrm{2}+\frac{\mathrm{1}}{\mathrm{x}}−\frac{\mathrm{2}}{\mathrm{x}^{\mathrm{3}} }}\right\}^{\mathrm{x}} \\ $$$$\Rightarrow\mathrm{f}\left(\mathrm{x}\right)\:=\mathrm{e}^{\mathrm{xln}\left(\frac{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }}{\mathrm{2}+\frac{\mathrm{1}}{\mathrm{x}}−\frac{\mathrm{2}}{\mathrm{x}^{\mathrm{3}} }}\right)} \:\sim\:\mathrm{e}^{−\mathrm{xln2}} \:\Rightarrow\mathrm{lim}_{\mathrm{x}\rightarrow+\infty} \mathrm{f}\left(\mathrm{x}\right)\:=\mathrm{0}\:\mathrm{and} \\ $$$$\mathrm{lim}_{\mathrm{x}\rightarrow−\infty} \mathrm{f}\left(\mathrm{x}\right)\:=+\infty \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com