Mathematics

When a rational function's limit exists and is finite and non-zero as $x \to \infty$, the numerator and denominator must grow at exactly the same rate—meaning they need the same degree. The leading terms will then determine the limit's value. Here, you need the highest power of $x$ in the numerator to match the highest power in the denominator, so the $x^m$ term competes with the $x^5$ term.

When a rational function's limit exists and is finite and non-zero as $x \to \infty$, the numerator and denominator must grow at exactly the same rate—meaning they need the same degree. The leading terms will then determine the limit's value. Here, you need the highest power of $x$ in the numerator to match the highest power in the denominator, so the $x^m$ term competes with the $x^5$ term.