What is a Limit?
A limit describes what value a function approaches as its input gets closer and closer to some value , without necessarily reaching it. Limits are the foundation of calculus: every derivative and every integral is defined using a limit.
In this lesson
1 The Intuition Behind Limits
Here is a weird but useful thought experiment. Imagine walking toward a wall, halving the distance with every step. You get closer and closer but in this version of reality you never actually touch it. A limit captures exactly that idea: what value are you approaching, even if you never arrive? , halving the distance each step. You approach the wall, but in this thought experiment you never actually touch it. A limit describes that approach: what value are you getting closer to, regardless of whether you arrive?
Take the function f(x) = (x² − 1)/(x − 1). Plug in x = 1 and you get 0/0, which is undefined. So the function has a hole at x = 1. But factor the top: (x+1)(x-1)/(x-1) = x+1 when x is not 1. So as x gets close to 1 from either side, f(x) gets close to 2. The hole exists, but the behavior around it is perfectly clear. But what happens as x gets close to 1? Factoring: (x²−1)/(x−1) = (x+1)(x−1)/(x−1) = x+1 (when x ≠ 1). So as x → 1, f(x) → 2. The function has a "hole" at x=1 but approaches 2 from both sides.
The limit of f(x) as x→a describes what f(x) approaches, not what f(a) equals. The function doesn't need to be defined at x=a for the limit to exist. This distinction , approaching vs arriving , is what makes limits powerful.
2 Limit Notation
lim(x→a) f(x) = L reads as "the limit of f(x) as x approaches a equals L." The symbol does a lot of work. It says: as x gets arbitrarily close to a from either side, f(x) gets arbitrarily close to L. Not at x = a. Getting close to a.
This means: as x gets arbitrarily close to a (from either side), f(x) gets arbitrarily close to L. The formal ε-δ definition makes "arbitrarily close" precise, but the intuition is sufficient for most applications.
6233 One-Sided Limits
Sometimes a function approaches different values from the left and right. One-sided limits capture this:
lim(x→a⁻) f(x) , limit from the left (x approaches a from values less than a)
lim(x→a⁺) f(x) , limit from the right (x approaches a from values greater than a)
A two-sided limit exists only when both one-sided limits exist and are equal. If lim(x→a⁻) f(x) ≠ lim(x→a⁺) f(x), the limit does not exist (DNE).
f(x) = |x|/x. From the right (x→0⁺): positive values divided by positive = +1. From the left (x→0⁻): negative values divided by negative = +1... wait, negative/negative = +1. Actually: from left, x is negative so |x| = −x, giving −x/x = −1. So lim(x→0⁻) = −1 and lim(x→0⁺) = +1. These differ, so lim(x→0) does not exist.
4 Infinite Limits and Asymptotes
When f(x) grows without bound as x→a, we write lim(x→a) f(x) = ∞. This doesn't mean the limit "equals infinity" , infinity isn't a number , it means the function increases without bound.
Example: lim(x→0) 1/x² = ∞. As x approaches 0, 1/x² grows arbitrarily large. The line x=0 is a vertical asymptote.
Limits at infinity: lim(x→∞) f(x) = L means f(x) approaches L as x grows without bound. lim(x→∞) 1/x = 0. The x-axis (y=0) is a horizontal asymptote of y = 1/x.
5 Why Limits Are the Foundation of Calculus
The derivative is defined as a limit: f'(x) = lim(h→0) [f(x+h) - f(x)] / h. That expression is the slope of the line between two nearby points on a curve. The limit shrinks the gap between them to zero, giving you the slope at a single point. , the limit of the slope of a secant line as the two points get infinitely close. Without limits, derivatives cannot be defined.
The definite integral is also a limit: the limit of a sum of infinitely many infinitely thin rectangles under a curve. Without limits, integration cannot be defined.
Every theorem in calculus , the chain rule, the mean value theorem, Taylor series , rests on the concept of a limit. Understanding limits deeply makes every subsequent calculus topic easier to grasp.
Practice Problems
Sources & Further Reading
The explanations on this page draw on the following established sources. We link to primary and secondary sources so you can verify claims and go deeper on any topic.