Limits at Infinity

Instead of $x$ approaching a finite number, we may ask what $f(x)$ approaches as $x$ grows without bound — written $\displaystyle\lim_{x\to\infty} f(x)$ (and similarly $x\to-\infty$). The single most useful fact is

because a fixed number divided by an ever-larger denominator shrinks to $0$.

The standard technique for a rational function $\dfrac{p(x)}{q(x)}$ as $x\to\infty$: divide every term of the numerator and denominator by the highest power of $x$ present in the denominator, then send each $\tfrac{1}{x^k}$ term to $0$.

Illustration. $\displaystyle\lim_{x\to\infty}\frac{3x^2 + 5x - 1}{2x^2 - x + 4}$. The highest power in the denominator is $x^2$; dividing every term by $x^2$:

The degree shortcut. For $\dfrac{p(x)}{q(x)}$ with numerator degree $m$ and denominator degree $n$:

• if $m < n$, the limit is $0$;
• if $m = n$, the limit is the ratio of the leading coefficients;
• if $m > n$, the expression grows without bound (limit is $\infty$ or $-\infty$).