01 Key Concepts
What 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.
One-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.
Limits at Infinity
Describe the behavior of a function as x grows without bound in the positive or negative direction — useful for finding horizontal asymptotes.
Indeterminate Forms
Expressions like 0/0 or infinity/infinity that need extra work (factoring, simplifying) before a limit can be evaluated directly.
Limit 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.
02 Key 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
03 Solved Examples
- Since 2x+1 is a simple polynomial, substitute x=3 directly.
- 2(3) + 1 = 6 + 1.
- Direct substitution gives 0/0, an indeterminate form.
- Factor the numerator: x^2 - 4 = (x-2)(x+2).
- Cancel the common (x-2) factor: (x-2)(x+2)/(x-2) = x+2.
- Now substitute x=2: 2+2.
- Divide every term by x: (3x/x) + (1/x) = 3 + 1/x.
- As x approaches infinity, 1/x approaches 0.
04 Practice Questions
📄 Limits — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.