Limits of $f(x)=x!\frac{e^{x}+1}{x^{x}}$? I have to say that I've no idea, but I just want to calculate $\lim\limits_{x\to\infty}f(x)$ 
$\lim\limits_{x\to\infty}\bigl(x!\dfrac{e^{x}+1}{x^{x}}\bigr)$
Use the Stirling approximation: $n! \approx \sqrt{2\pi n} \frac{n^n}{e^n}$. Thus, $x! \frac{e^x+1}{x^x} \approx \sqrt{2\pi x} \frac{e^x+1}{e^x} \approx \sqrt{2\pi x}$ for large $x$. If you meant $e^{x+1}$ instead, you get $x! \frac{e^{x+1}}{x^x} \approx \sqrt{2\pi x} e$.
The limit of $f(x)=x!\frac{e^{x}+1}{x^{x}}$ when $x\to\infty$ is $0$.