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lim-x-3x-12-6x-20-2x-




Question Number 149346 by Study last updated on 04/Aug/21
lim_(x→∞) (((3x+12)/(6x−20)))^(2x) =?
$${li}\underset{{x}\rightarrow\infty} {{m}}\left(\frac{\mathrm{3}{x}+\mathrm{12}}{\mathrm{6}{x}−\mathrm{20}}\right)^{\mathrm{2}{x}} =? \\ $$
Commented by Study last updated on 04/Aug/21
helpe me
$${helpe}\:{me} \\ $$
Answered by gsk2684 last updated on 04/Aug/21
lim_(x→∞) ((3x+12)/(6x−20))=lim_(x→∞) ((3+((12)/x))/(6−((20)/x)))=((3+0)/(6−0))=(1/2)<1  then  lim_(x→∞) (((3x+12)/(6x−20)))^(2x) =((1/2))^∞ =0
$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{3}{x}+\mathrm{12}}{\mathrm{6}{x}−\mathrm{20}}=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{3}+\frac{\mathrm{12}}{{x}}}{\mathrm{6}−\frac{\mathrm{20}}{{x}}}=\frac{\mathrm{3}+\mathrm{0}}{\mathrm{6}−\mathrm{0}}=\frac{\mathrm{1}}{\mathrm{2}}<\mathrm{1} \\ $$$${then} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{3}{x}+\mathrm{12}}{\mathrm{6}{x}−\mathrm{20}}\right)^{\mathrm{2}{x}} =\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\infty} =\mathrm{0} \\ $$
Answered by puissant last updated on 04/Aug/21
=lim_(x→∞) e^(2xln(((3x+12)/(6x−20))))   =lim_(x→∞) e^(2xln(((3+((12)/x))/(6−((20)/x)))))   =lim_(x→∞) e^(2xln((1/2)))   =lim_(x→∞ ) e^(−2xln2)   = { ((+∞    if   x→−∞)),((0    if    x→+∞)) :}
$$=\mathrm{lim}_{\mathrm{x}\rightarrow\infty} \mathrm{e}^{\mathrm{2xln}\left(\frac{\mathrm{3x}+\mathrm{12}}{\mathrm{6x}−\mathrm{20}}\right)} \\ $$$$=\mathrm{lim}_{\mathrm{x}\rightarrow\infty} \mathrm{e}^{\mathrm{2xln}\left(\frac{\mathrm{3}+\frac{\mathrm{12}}{\mathrm{x}}}{\mathrm{6}−\frac{\mathrm{20}}{\mathrm{x}}}\right)} \\ $$$$=\mathrm{lim}_{\mathrm{x}\rightarrow\infty} \mathrm{e}^{\mathrm{2xln}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)} \\ $$$$=\mathrm{lim}_{\mathrm{x}\rightarrow\infty\:} \mathrm{e}^{−\mathrm{2xln2}} \\ $$$$=\begin{cases}{+\infty\:\:\:\:\mathrm{if}\:\:\:\mathrm{x}\rightarrow−\infty}\\{\mathrm{0}\:\:\:\:\mathrm{if}\:\:\:\:\mathrm{x}\rightarrow+\infty}\end{cases} \\ $$

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