Limit at infinity and horizontal asymptote

As $x \to \pm\infty$, the denominator $x^2$ grows without bound, so the fraction $\dfrac{2}{x^2}$ approaches $0$. This is true regardless of whether $x$ approaches $+\infty$ or $-\infty$, since $x^2$ is always positive and grows large in either direction.

Therefore $g(x) = 5 - \dfrac{2}{x^2} \to 5 - 0 = 5$ as $x \to \infty$ and as $x \to -\infty$. Since both limits equal the finite value $5$, the graph of $g(x)$ has a horizontal asymptote at $y = 5$.