End behavior of a rational function with numerator degree greater than denominator degree

Here the numerator $2x^3 - x$ has degree 3, while the denominator $x^2+5$ has degree 2. Since the degree of the numerator is greater than the degree of the denominator, the rational function grows without bound as $x\to\infty$, so the limit is not a finite number.

To see this explicitly, divide numerator and denominator by $x^2$: $g(x) = \dfrac{2x - 1/x}{1 + 5/x^2}$. As $x \to \infty$, the terms $1/x$ and $5/x^2$ vanish, leaving $g(x) \approx 2x$, which grows to $+\infty$. Therefore $\lim_{x\to\infty} g(x) = +\infty$, and since the limit is infinite (not a finite value $L$), $g(x)$ has no horizontal asymptote.