Calculus · Quick Reference

Limits Cheat Sheet

Differential Calculus · Lesson 1/20

In one line: a limit describes the value a function approaches as its input gets closer and closer to some point — even if the function is never actually defined at that point.

Key Ideas

1What a Limit Means. lim as x approaches a of f(x) = L means f(x) gets arbitrarily close to L as x gets close to a, from either side.

2One-Sided Limits. The left-hand limit (x approaches a from below) and right-hand limit (x approaches a from above) can differ. The limit only exists if both sides agree.

3Limits at Infinity. Describe the behavior of a function as x grows without bound in the positive or negative direction — useful for finding horizontal asymptotes.

4Indeterminate Forms. Expressions like 0/0 or infinity/infinity that need extra work (factoring, simplifying) before a limit can be evaluated directly.

5Limit Laws. Limits of sums, differences, products, and quotients can be found by taking the limit of each piece separately, as long as the results are defined.

Worked Examples

Find lim as x approaches 3 of (2x + 1).

Find lim as x approaches 2 of (x^2 - 4)/(x - 2).

Find lim as x approaches infinity of (3x + 1)/(x).

Formulas

lim (x->a) [f(x) + g(x)] = lim f(x) + lim g(x)

lim (x->a) [f(x) * g(x)] = lim f(x) * lim g(x)

lim (x->a) c = c, for any constant c

Practice Yourself

Find lim as x approaches 1 of (3x + 2).

Find lim as x approaches 0 of (x^2 + 5).

Find lim as x approaches 5 of (x^2-25)/(x-5).