1. **Problem Statement:** Determine if the function $f(x) = x^4 - x^2$ has a horizontal asymptote. 2. **Recall the definition:** A horizontal asymptote is a horizontal line $y = L$ where $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$. 3. **Calculate the limits:** $$\lim_{x \to \infty} (x^4 - x^2)$$ As $x \to \infty$, $x^4$ grows much faster than $x^2$, so the dominant term is $x^4$. Therefore, $$\lim_{x \to \infty} (x^4 - x^2) = \infty$$ 4. Similarly, for $x \to -\infty$: $$\lim_{x \to -\infty} (x^4 - x^2)$$ Since $x^4$ is even power, it is positive and grows large as $x \to -\infty$, and $x^2$ is also positive. Thus, $$\lim_{x \to -\infty} (x^4 - x^2) = \infty$$ 5. **Conclusion:** Since both limits go to infinity, the function does not approach a finite constant value at infinity or negative infinity. Therefore, the function $f(x) = x^4 - x^2$ has **no horizontal asymptote**.