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Question Number 184341 by Davidtim last updated on 05/Jan/23

lim_(x→∞) (x^2 /3^x )=?

$${li}\underset{{x}\rightarrow\infty} {{m}}\frac{{x}^{\mathrm{2}} }{\mathrm{3}^{{x}} }=? \\ $$

Commented by JDamian last updated on 05/Jan/23

zero, as the denominator tends to infinity faster than the numerator does.

Answered by SEKRET last updated on 05/Jan/23

   lim_(x→∞)  (x^2 /3^x )= lim_(x→∞)  ((2x)/(3^x ln(3)))=lim_(x→∞) (2/(3^x ln^2 (3)))=0

$$\:\:\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\infty} {\boldsymbol{\mathrm{lim}}}\:\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }{\mathrm{3}^{\boldsymbol{\mathrm{x}}} }=\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\infty} {\boldsymbol{\mathrm{lim}}}\:\frac{\mathrm{2}\boldsymbol{\mathrm{x}}}{\mathrm{3}^{\boldsymbol{\mathrm{x}}} \boldsymbol{\mathrm{ln}}\left(\mathrm{3}\right)}=\underset{\boldsymbol{\mathrm{x}}\rightarrow\infty} {\boldsymbol{\mathrm{lim}}}\frac{\mathrm{2}}{\mathrm{3}^{\boldsymbol{\mathrm{x}}} \boldsymbol{\mathrm{ln}}^{\mathrm{2}} \left(\mathrm{3}\right)}=\mathrm{0} \\ $$

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