Related facts about the natural logarithm

From the preceding it is easy to see that the natural logarithm function grows more slowly than any positive power function. For example, to investigate

$$\frac{\ln(u)}{u}$$

as u tends to infinity, substitute u = e^x to get

$$\frac{\ln(u)}{u} = \frac{\ln(e^x)}{e^x} = \frac{x}{e^x}$$

Since u tends to infinity is the same as x tends to infinity, we see that

$$\lim_{u\rightarrow+\infty}\frac{\ln(u)}{u} = 0.$$

We can also investigate the behaviour of $u\ln(u)$ as u tends to through the positive numbers by substituting u = e^-x to get

$$u\ln(u) = e^{-x}\ln(e^{-x}) = -\frac{x}{e^x}$$

Since u tends to through the positive numbers is the same as x tends to $+\infty$, we see

$$\lim_{u\rightarrow 0^+}u\ln(u) = 0.$$