How to attack a limit
- Substitute first. If you get a number, that's the answer.
- 0/0 or ∞/∞: factorise and cancel, rationalise a square root, or reshape it into one of the standard limits above. L'Hôpital's rule (differentiate top and bottom) also works.
- 1^∞: use lim f^g = e^(lim g(f − 1)).
- x → ∞ with polynomials: divide top and bottom by the highest power of x.
Questions students ask
What is lim x→0 sin x/x?
1, with x in radians.
What is lim x→0 (1 + x)^(1/x)?
e.
What is lim x→0 (aˣ − 1)/x?
log a (natural log).
When can I use L'Hôpital's rule?
Only for 0/0 or ∞/∞ forms. Differentiate the top and bottom separately, then take the limit again.