A short and non rigorous explanation: in a fraction $\dfrac{P(x)}{Q(x)}$ (assume that both $P(x)$ and $Q(x)$ are positive) it may happen that $P(x)$ goes to infinity "faster", "slower" or "as fast as" $Q(x)$. Depending on these speeds the fraction tends to infinity, zero or a constant, or it may happen that the limit does not exist, as commented by Samuel. To find the limit one can apply L'Hôpital's rule, if $P(x)$ and $Q(x)$ are differentiable.
Both $\lim\limits_{x \to \infty} \frac{x}{x}$ and $\lim\limits_{x \to \infty} \frac{2^x}{2^x}$ is $1$.