\(\displaystyle\lim_{x\to\infty}(5\e^{-x}+1)^2e^x=\lim_{x\to\infty}\frac{(5e^{-x}+1)^2e^{2x}}{e^x}=\lim_{x\to\infty}\frac{(5+e^x)^2}{e^x}\)
Replacing \(y=e^x\), the limit is equivalent to
\(\displaystyle\lim_{y\to\infty}\frac{(5+y)^2}y\)
which clearly approaches \(\infty\).
