l'Hôpital vs Other Methods

$$\lim_{x \rightarrow 0} \frac{x^4}{x^4+x^2} = \lim_{x \rightarrow 0} \frac{\frac{d}{dx}(x^4)}{\frac{d}{dx}(x^4+x^2)} = \lim_{x \rightarrow 0} \frac{4x^3}{4x^3+2x} = \lim_{x\to0} \frac{12x^2}{12x^2+2} = \frac{0}{0+2} = 0$$

There. You can't apply l'Hospital there because the denominator doesn't go to $0$.

You haven't checked whether L'Hopital could be applied each time.

After doing derivative one more time you get $12x^2 +2 $ which is not $0$ when $x$ goes to $0$.