Doeas $ \int_0^{0.5} \frac{\mathrm dx}{\sin x\ln x} $ exist?

Question

So I have a test next week and I found myself struggled in a question.

Does $ \ \int_{0}^{0.5} \frac{\mathrm dx}{\sin x\ln x} \ $ exist ?

So I saw an answer to that question which I do not understand why is it true.

It says that because $\frac{\sin x}{x} \rightarrow 1$ then we just need to check that $ \ \int_{0}^{0.5} \frac{\mathrm dx}{x\ln x } \ $ exists (which I calculated and that is not true).

But why is it true ?

If there is another way I will be happy to learn.

Thanks in advance !!

Answer 1

At $x=0$, the integrand behaves as $1/(x \log{x})$, which has integral

$$\int \frac{d(\log{x})}{\log{x}} = \log{\log{x}}$$

which blows up there. The integral thus diverges.