Math basics › More calculus

Limits at infinity

Compare the growth

For a ratio of polynomials, only the highest powers matter as x runs off. Bottom wins: the limit is 0. Equal: the ratio of leading coefficients. Top wins: it grows without limit.

What grows fastest

Exponentials beat powers, which beat logs. eˣ outruns x¹⁰⁰ eventually, and ln x loses to every positive power of x.

L'Hôpital's rule

If substituting gives 0/0 or ∞/∞, differentiate the top and bottom separately and try again. Not the quotient rule — each one on its own.

Only for those two forms

Using L'Hôpital on a limit that is not indeterminate gives the wrong answer. Substitute first, and only reach for it when you get 0/0 or ∞/∞.