1. **Problem Statement:** Find the asymptotes of the function $f(x) = \frac{x^2}{x^2 + 4}$.\n\n2. **Vertical Asymptotes:** These occur where the denominator is zero and the numerator is not zero. Here, the denominator is $x^2 + 4$, which is never zero for any real $x$ because $x^2 \geq 0$ and $4 > 0$. So, there are no vertical asymptotes.\n\n3. **Horizontal Asymptotes:** To find horizontal asymptotes, analyze the behavior of $f(x)$ as $x \to \pm \infty$.\n\nCalculate the limit:\n$$\lim_{x \to \pm \infty} \frac{x^2}{x^2 + 4} = \lim_{x \to \pm \infty} \frac{1}{1 + \frac{4}{x^2}} = 1.$$\n\nSo, the horizontal asymptote is $y = 1$.\n\n**Final answer:** The function has no vertical asymptotes and one horizontal asymptote at $y = 1$.