Evaluate the following limits. lim_{xrightarrowinfty}(3cdot2^{1-x}+x^2cdot2^{1-x})

Evaluate limit:
$\underset{x\to \mathrm{\infty }}{lim}(3\cdot 2^{1-x}+x^2\cdot 2^{1-x})$
$\underset{x\to a}{lim}[f(x)+-g(x)]=\underset{x\to a}{lim}f(x)+-\underset{x\to a}{lim}g(x)$
$\underset{x\to \mathrm{\infty }}{lim}(3\cdot 2^{1-x}+x^2\cdot 2^{1-x})=\underset{x\to \mathrm{\infty }}{lim}(3\cdot 2^{1-x})+\underset{x\to \mathrm{\infty }}{lim}(x^2\cdot 2^{1-x})$
$=3\cdot \underset{x\to \mathrm{\infty }}{lim}(2^{1-x})+2\cdot \underset{x\to \mathrm{\infty }}{lim}(x^2\cdot 2^{-x})$
$=3\cdot \underset{x\to \mathrm{\infty }}{lim}(2^{1-x})+2\cdot \underset{x\to \mathrm{\infty }}{lim}(x^2\cdot \frac{1}{2^x})$
Take $3\cdot \underset{x\to \mathrm{\infty }}{lim}(2^{1-x})$
Apply exponent rule $a^x=e^{\ln a^x}=e^{x\cdot \ln a}$
$=3\cdot \underset{x\to \mathrm{\infty }}{lim}(e^{(1-x)\cdot \ln 2})$
$=3\cdot \underset{x\to \mathrm{\infty }}{lim}{2}^{1-x}$
$=3\cdot \underset{x\to \mathrm{\infty }}{lim}{e}^{x\cdot \ln 2}$
Apple the Limit Chain Rule:
$g(x)=x\ln (2),f(u)=e^u$
$=-\mathrm{\infty }\cdot \ln 2$
$=-\mathrm{\infty }$
$3\cdot \underset{x\to \mathrm{\infty }}{lim}{e}^{x\cdot \ln 2}=3\cdot e^{-\mathrm{\infty }}$
$=3\cdot 0$
$=0$
Take $2\cdot \underset{x\to \mathrm{\infty }}{lim}{x}^2\cdot \frac{1}{2^x}$
if sum $a_n$ converges,then $\underset{n\to \mathrm{\infty }}{lim}(a_n)=0$
Apply ratio test and check weather series is convergent of divergent
if $|\frac{a_{n+1}}{a_n}|\le q$ eventually for some 0, then $\sum _{n=1}^x|a_n|$ converges,
if $|\frac{a_{n+1}}{a_n}|>1$ eventually then $\sum _{n=1}^x{a}_n$ diverges
$\underset{x\to \mathrm{\infty }}{lim}(\frac{x^2}{2^x})=0$
$=2\cdot 0$
$=0$
$\underset{x\to \mathrm{\infty }}{lim}(3\cdot 2^{1-x}+x^2\cdot 2^{1-x})=3\cdot \underset{x\to \mathrm{\infty }}{lim}(2^{1-x})+2\cdot \underset{x\to \mathrm{\infty }}{lim}(x^2\cdot \frac{1}{2^x})$
$=0+0$
$=0$
Result:
$3\cdot \underset{x\to \mathrm{\infty }}{lim}(2^{1-x})+2\cdot \underset{x\to \mathrm{\infty }}{lim}(x^2\cdot \frac{1}{2^x})=0$