How can I show the two limits How can I show the two limits
$$
\displaylines{
  \mathop {\lim }\limits_{x \to  + \infty } \frac{{x^2 e^{x + \frac{1}{x}} }}{{e^{ - x} \left( {\ln x} \right)^2 \sqrt x }} = \mathop {\lim }\limits_{x \to  + \infty } \left( {\frac{{x^{\frac{3}{4}} e^{\left( {x + \frac{1}{{2x}}} \right)} }}{{\ln x}}} \right)^2  =  + \infty  \cr 
  \mathop {\lim }\limits_{x \to 0^ +  } \left( {\frac{{x^{\frac{3}{4}} e^{\left( {x + \frac{1}{{2x}}} \right)} }}{{\ln x}}} \right)^2  =  + \infty  \cr}
$$
think you

$$\frac{x^2e^x}{e^{-x} (\ln x)^2\sqrt x} < \frac{x^2e^{x + \frac1x}}{e^{-x} (\ln x)^2\sqrt x} \text{ for large $x$ since } e^{\frac1x} > 1$$
$$$$
$$\lim_{x \to \infty} \frac{x^2e^x}{e^{-x} (\ln x)^2\sqrt x} = e^{2x}\frac{x^{1.5}}{(\ln x)^2} = +\infty$$
$$$$
$$\text{Thus, the right hand side at the top should diverge to $+\infty$ too.}$$
