Comput this limit withou L'Hospital's rule

Question

How can I find $\lim \limits_{x\to 0} \frac{x \sin^2 x}{(\arctan 5x)^2}$ without using l'Hopital's rule?

I understand that I can simplify the numerator but what should I do with the denominator?

Answer 1

Hint. Write $$ \frac{x \sin^2 x}{\arctan^2 5x} = \left(\frac{\arctan 5x}{5x}\right)^{-2} \cdot \left(\frac{\sin x}x\right)^2 \cdot \frac{x \cdot x^2}{25 x^2} $$ Now use the (hopefully known) limits of $\frac{\arctan x}{x}$ and $\frac{\sin x}x$ for $x \to 0$.