Chapter 4.3, Problem 55ES

Using $\text{L'H <mover accent="true"> o <mo> ^ </mo> </mover>pital's}$ rule (Section 3.6) one can verify that $\underset{x\to +\infty }{\lim }\frac{\ln x}{x^r}=0,\underset{x\to +\infty }{\lim }\frac{x^r}{\ln x}=+\infty ,\underset{x\to 0^{+}}{\lim }{x}^r\ln x=0$

for any positive real number $r$ . In these exercises: (a) Use these results, as necessary, to find the limits of $f\left(x\right)$ as $x\to +\infty$ and as $x\to 0^{+}$ . (b) Sketch a graph of $f\left(x\right)$ and identify all relative extrema, inflection points, and asymptotes (as appropriate). Check your work with a graphing utility.

$f\left(x\right)=x\ln x$