Definition of a limit
Definition of a Limit
Definition of a Limit is the opening topic of Limits and Continuity in the California Calculus Standards. It is aligned to Standard 1.0, which requires students to demonstrate knowledge of both the formal definition and the graphical interpretation of the limit of a function, including one-sided limits.
A limit is the value a function \(f(x)\) approaches as \(x\) approaches a fixed number \(a\), written \(\lim_{x\to a} f(x)=L\). Mastering this definition — left-hand and right-hand limits, when a two-sided limit exists, and when a limit does not exist — is the foundation for continuity, the derivative, and the integral later in the course.
Theory
A limit describes the value a function approaches as its input gets close to a fixed number, written \(\lim_{x\to a} f(x)=L\). This page explains one-sided limits, when a two-sided limit exists, how to read a limit from a graph, how to evaluate one by direct substitution, and when a limit does not exist.
A limit is the single value a function heads toward as \(x\) gets close to a fixed number \(a\). We write
which reads “the limit of \(f(x)\), as \(x\) approaches \(a\), equals \(L\).” It means \(f(x)\) gets as close as we like to \(L\) when \(x\) is close to \(a\) — on both sides of \(a\).
A one-sided limit looks at only one direction. The left-hand limit \(\lim_{x\to a^{-}} f(x)\) uses values with \(x
The limit notation and the test for existence:
The one-sided limits (from the left and from the right):
The two-sided limit exists exactly when they agree:
How to find a limit
- Substitute \(x=a\) into \(f(x)\). If you get a single number, that is the limit.
- Simplify if substitution gives the indeterminate form \(\dfrac{0}{0}\): factor the top and bottom, cancel the common factor, then substitute again.
- Check both sides when the graph has a break at \(a\): find \(\lim_{x\to a^{-}}\) and \(\lim_{x\to a^{+}}\). If they are equal, that common value is the limit; if they differ, the limit does not exist.
The function is a polynomial, so substitute \(x=2\).
| \(\lim_{x\to 2}(3x+1)\) | \(=\) | \(3\cdot 2+1\) |
| \(=\) | \(6+1\) | |
| \(=\) | \(7\) |
Substituting gives \(\dfrac{0}{0}\), so factor and cancel first.
| \(\dfrac{x^2-16}{x-4}\) | \(=\) | \(\dfrac{(x-4)(x+4)}{x-4}\) |
| \(=\) | \(x+4\) |
Now substitute \(x=4\):
| \(\lim_{x\to 4}(x+4)\) | \(=\) | \(4+4=8\) |
| \(\lim_{x\to 1^{-}} g(x)\) | \(=\) | \(2\) |
| \(\lim_{x\to 1^{+}} g(x)\) | \(=\) | \(3\) |
The one-sided limits differ (\(2\neq 3\)), so \(\lim_{x\to 1} g(x)\) does not exist.
The limit follows the curve, which heads to \(4\) from both sides. The plotted value \(f(1)=1\) does not matter.
| \(\lim_{x\to 1} f(x)\) | \(=\) | \(4\) |
| \(f(1)\) | \(=\) | \(1\) |
The limit is \(4\), even though \(f(1)=1\).
Common pitfalls
Frequently asked questions
What is a limit in calculus?
The single value a function heads toward as \(x\) gets close to a fixed number \(a\). Formally, \(\lim_{x\to a} f(x)=L\) means \(f(x)\) gets arbitrarily close to \(L\) as \(x\) approaches \(a\) from both sides.
How do you find a limit?
Try direct substitution first. If you get a number, that is the limit. If you get \(\dfrac{0}{0}\), factor and cancel, then substitute. If the left and right sides disagree, the limit does not exist.
Does the function have to be defined at the point?
No. The limit depends only on values near \(a\), not on \(f(a)\). A function can have a hole at \(x=a\) and still have a limit there.
What does it mean when a limit does not exist?
Most often the left-hand and right-hand limits are different numbers — a jump in the graph. If the two sides do not agree on one value, the two-sided limit does not exist.
What is the difference between a one-sided and a two-sided limit?
A one-sided limit approaches \(a\) from only the left or only the right. The two-sided limit exists only when both one-sided limits are equal.