A limit is the value a function gets closer and closer to as the input approaches a certain number. We write limx → af(x) = L. The function need not even be defined at a. For f(x) = x2 − 1x − 1, f(1) is undefined, but the limit as x → 1 is 2.
In Spanish
límite
Also called
limit of a function, lim, limiting value
Grade level
Calculus
PrecalculusCalculusAP
Limit in a picture
The function has a hole at x = 1, yet from both sides its values close in on 2. The limit is about the approach, not the point itself.
Formal definition and notation
limx → af(x) = L means that f(x) can be made as close to L as we like by taking x close enough to a, with x ≠ a. The limit exists only when the left-hand limit (x → a−) and the right-hand limit (x → a+) both exist and are equal. The precise version uses epsilon and delta distances.
Symbol
Read it as
Meaning
limx → af(x) = L
the limit of f of x as x approaches a equals L
f(x) heads toward L as x heads toward a
x → a−
x approaches a from the left
only x-values less than a
x → a+
x approaches a from the right
only x-values greater than a
x → ∞
x approaches infinity
x grows without bound; the limit describes long-run behavior
Key facts
A limit describes where the function is heading, not where it is: f(a) and the limit can be different, or f(a) can be undefined.
For polynomials, the limit is just the value: limx → 2 (x2 + 1) = 5.
The limit does not exist if the left and right sides head to different values (a jump) or the function grows without bound (a vertical asymptote).
Limits at infinity give horizontal asymptotes: limx → ∞1x = 0.
Limits are the foundation of calculus: both the derivative and the integral are defined as limits.
How to find a limit
How to find a limit
limx → af(x) = f(a) when f is continuous at a
a
the input value being approached
f(a)
the value you get by direct substitution
Try direct substitution first. If you get 00, factor and cancel, rationalize, or check a table and the graph. 00 does not mean the limit is 0 or undefined; it means more work is needed.
Limit examples, step by step
Example 1: Direct substitution
Warm-up
Find limx → 3 (2x + 1).
2x + 1 is a polynomial, so it has no holes or jumps.
A derivative is one particular limit: the limit of the slopes of secant lines as the two points come together.
Quick check: 3 questions
Question 1 of 3
What is limx → 4 (x2 − 3x)?
Show answer and explanation
Answer: A. A polynomial is continuous, so substitute: 16 − 12 = 4. 16 forgets the −3x term, and 28 adds it.
Question 2 of 3
What is limx → −1x2 − 1x + 1?
Show answer and explanation
Answer: C. Factor: (x − 1)(x + 1)x + 1 = x − 1 for x ≠ −1, so the limit is −1 − 1 = −2. 00 does not mean 0 or does not exist.
Question 3 of 3
As x → 0−, f(x) → 5; as x → 0+, f(x) → 1. What is limx → 0f(x)?
Show answer and explanation
Answer: D. The one-sided limits disagree, so the two-sided limit does not exist. 3 is the average of the two sides, which is not how limits work.
Limit: frequently asked questions
Can a limit exist if the function is undefined at that point?
Yes. A limit only looks at values near the point. f(x) = x2 − 9x − 3 is undefined at x = 3, but for every other x it equals x + 3, so the limit as x → 3 is 6. The graph is a line with a hole at (3, 6).
When does a limit not exist?
Three common cases. The left and right sides approach different values (a jump). The function grows without bound, as 1x2 does near 0, where we write the limit as infinity but it does not exist as a number. Or the function oscillates forever, as the sine of 1x does near 0.
Why do we need limits?
Calculus asks questions that ordinary arithmetic cannot answer, such as the slope of a curve at a single point, which seems to need dividing 0 by 0. Limits get around this by looking at what happens as two points get closer and closer. Every derivative and every integral is defined as a limit.
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