Limits at infinity

As $x$ becomes huge, small terms stop mattering. In $3 x^{2} + 5 x + 1$ , the $3 x^{2}$ dominates. So for ratios, only the highest powers matter.

Comparison	Limit	Example
Bottom's power is bigger	0	$\frac{x}{x ^{2}}$
Powers equal	Ratio of leading numbers	$\frac{3 x ^{2}}{2 x ^{2}}$ gives $\frac{3}{2}$
Top's power is bigger	$\pm \infty$	$\frac{x ^{2}}{x}$

Work through it

$lim_{x \to \infty} \frac{3 x ^{2} + 5}{2 x ^{2} - x}$

Divide every term by $x^{2}$ . $\frac{3 + 5/ x ^{2}}{2 - 1/ x}$

As $x$ grows, those fractions vanish. $\frac{3}{2}$

$lim_{x \to \infty} \frac{4 x + 1}{x ^{2}}$

The bottom grows faster. $0$

Watch out for this

Saying $\frac{\infty}{\infty} = 1$ .

That's not a calculation. It depends on how fast each one grows.

The short version
For a ratio of polynomials, compare the highest powers. Bigger bottom gives 0, equal gives the ratio of leading numbers, and bigger top gives infinity.

Your turn

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