Use L'Hôpital's rule to solve $\lim_{x\to 0^{+}}\sin(x)\ln(x)$

Question

Use L'Hôpital's rule to solve

$$\lim_{x\to 0^{+}}\sin(x)\ln(x)$$

My attempt:

$$\lim_{x\to 0^{+}}\sin(x)\ln(x) = \lim_{x\to 0^{+}} \frac{\ln(x)}{\sin(x)}$$

$\frac{\ln(0)}{\sin(0)}$ is in the form $\frac{-\infty}{0}$, it is indeterminate, and as such, using L'Hôpital's rule:

$$\lim_{x\to 0^{+}} \frac{\ln(x)}{\sin(x)} = \lim_{x\to0^{+}}\frac{1}{x\cos(x)}$$

I would have applied L'Hôpital's rule again, but to my horror, I realise that $\frac{1}{0}$ is not an indeterminate form, according to Wikipedia.

I realized that my reasoning, while it could let me get the correct answer, is wrong! How do you solve this question now?

EDIT: Is $\frac{-\infty}{0}$ indeterminate as well? I couldn't find it in Wikipedia. If it is not indeterminate, I couldn't use L'Hôpital's rule too!

Answer 1

It would be re-written as : $$\lim_{x \to 0^+} \frac{\ln (x)}{\csc (x)}$$

Instead of $$ \frac{\ln(x)}{\sin(x)}$$ Now use L'Hopital's rule.

Also note that the form $\dfrac{\infty}{0}$ isn't indeterminate, it already tends to $\infty$. The problem in your solution is that you accidentally wrote : $$\frac{1}{\sin x}= \sin x$$

Answer 2

$$\lim_{x\to0^{+}} \frac{\ln(x)}{\csc(x)} = \lim_{x\to0^{+}} \frac{-1}{x\csc(x)\cot(x)} = \lim_{x\to0^{+}} \frac{-\sin(x)}{x} \lim_{x\to0^{+}} \tan(x) = 0$$