Let

1. Show that

   $$f^{\prime }(x)=f(x)(1-\frac{n}{x})$$

   and

   $$f^{″}(x)=f(x)(1-\frac{2n}{x}+\frac{n(n+1)}{x^2})$$

2. Find the range of values of $x$ for which $f(x)$ is increasing.

3. Show that $f^{\prime }(x)$ is increasing for all values of $x>0.$

4. Use the above work to explain why

   $$\frac{e^x}{x^n}\to \mathrm{\infty }\text{ as }x\to \mathrm{\infty }$$

   (This problem shows that exponential functions grow much more quickly than polynomials!)