Limit Formula

Limit is the value a function gets closer and closer to as the input approaches a specific target value, without necessarily reaching it.

The Formula

limโกxโ†’af(x)=Lโ€…โ€ŠโŸบโ€…โ€Šโˆ€ฯต>0,โ€…โ€Šโˆƒฮด>0ย s.t.ย 0<โˆฃxโˆ’aโˆฃ<ฮดโ‡’โˆฃf(x)โˆ’Lโˆฃ<ฯต\lim_{x \to a} f(x) = L \iff \forall \epsilon > 0,\; \exists \delta > 0 \text{ s.t. } 0 < |x - a| < \delta \Rightarrow |f(x) - L| < \epsilon

When to use: What output do you get closer and closer to as you get closer and closer to some input?

Quick Example

limโกxโ†’2x2=4\lim_{x \to 2} x^2 = 4 As xx gets closer to 2, x2x^2 gets closer to 4.

Notation

limโกxโ†’af(x)=L\lim_{x \to a} f(x) = L

What This Formula Means

The value a function gets closer and closer to as the input approaches a specific target value, without necessarily reaching it.

What output do you get closer and closer to as you get closer and closer to some input?

Formal View

limโกxโ†’af(x)=Lโ€…โ€ŠโŸบโ€…โ€Šโˆ€ฯต>0,โ€…โ€Šโˆƒฮด>0:0<โˆฃxโˆ’aโˆฃ<ฮดโ€…โ€ŠโŸนโ€…โ€Šโˆฃf(x)โˆ’Lโˆฃ<ฯต\lim_{x \to a} f(x) = L \iff \forall \epsilon > 0,\; \exists \delta > 0 : 0 < |x - a| < \delta \implies |f(x) - L| < \epsilon

Worked Examples

Example 1

easy
Find limโกxโ†’3(2x+1)\lim_{x \to 3} (2x + 1)

Answer

limโกxโ†’3(2x+1)=7\lim_{x \to 3} (2x + 1) = 7

First step

1
Since 2x+12x + 1 is a polynomial, it is continuous everywhere.

Full solution

  1. 2
    For continuous functions, we can evaluate the limit by direct substitution.
  2. 3
    Substitute x=3x = 3: 2(3)+1=6+1=72(3) + 1 = 6 + 1 = 7.
When a function is continuous at a point, the limit equals the function value at that point. Polynomials are continuous everywhere, so direct substitution always works for polynomial limits.

Example 2

medium
Find limโกxโ†’2x2โˆ’4xโˆ’2\lim_{x \to 2} \frac{x^2 - 4}{x - 2}

Example 3

hard
Find limโกxโ†’0sinโกxx\lim_{x \to 0} \frac{\sin x}{x}

Common Mistakes

  • Just substituting aa and stopping when you get 00\frac{0}{0} โ€” that's an indeterminate form, so factor, cancel, or rationalize first to reveal the real limit.
  • Assuming the limit fails to exist because f(a)f(a) is undefined โ€” the limit depends only on nearby values, not the value at aa.
  • Ignoring that the left and right approaches must agree โ€” if they give different values, the two-sided limit does not exist.

Why This Formula Matters

The limit is the foundation every other calculus idea is built on: derivatives are limits of slopes and integrals are limits of sums. Students who treat limโกxโ†’af(x)\lim_{x\to a}f(x) as just 'plug in aa' break the moment they meet a 00\frac{0}{0} form like x2โˆ’1xโˆ’1\frac{x^2-1}{x-1}, where the function has a hole but the limit is perfectly real. Recognizing it by "Am I asked what the output heads toward as the input closes in, rather than the output exactly at that input?" โ€” rather than by familiar numbers โ€” is what lets a student tell it apart from function value f(a)f(a) and continuity and derivative in a mixed problem set.

Frequently Asked Questions

What is the Limit formula?

The value a function gets closer and closer to as the input approaches a specific target value, without necessarily reaching it.

How do you use the Limit formula?

What output do you get closer and closer to as you get closer and closer to some input?

What do the symbols mean in the Limit formula?

limโกxโ†’af(x)=L\lim_{x \to a} f(x) = L

Why is the Limit formula important in Math?

The limit is the foundation every other calculus idea is built on: derivatives are limits of slopes and integrals are limits of sums. Students who treat limโกxโ†’af(x)\lim_{x\to a}f(x) as just 'plug in aa' break the moment they meet a 00\frac{0}{0} form like x2โˆ’1xโˆ’1\frac{x^2-1}{x-1}, where the function has a hole but the limit is perfectly real. Recognizing it by "Am I asked what the output heads toward as the input closes in, rather than the output exactly at that input?" โ€” rather than by familiar numbers โ€” is what lets a student tell it apart from function value f(a)f(a) and continuity and derivative in a mixed problem set.

What do students get wrong about Limit?

The procedure for limit is the easy part; the trap is just substituting aa and stopping when you get 00\frac{0}{0}. Asking "Am I asked what the output heads toward as the input closes in, rather than the output exactly at that input?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Limit formula?

Before studying the Limit formula, you should understand: function definition.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Limits Explained Intuitively: The Foundation of Calculus โ†’
โ† Back to Limit See All Examples Practice Problems