Limit Formula

The Formula

\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

\lim_{x \to 2} x^2 = 4 As x gets closer to 2, x^2 gets closer to 4.

Notation

\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

\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 \lim_{x \to 3} (2x + 1)

Solution

  1. 1
    Since 2x + 1 is a polynomial, it is continuous everywhere.
  2. 2
    For continuous functions, we can evaluate the limit by direct substitution.
  3. 3
    Substitute x = 3: 2(3) + 1 = 6 + 1 = 7.

Answer

\lim_{x \to 3} (2x + 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 \lim_{x \to 2} \frac{x^2 - 4}{x - 2}

Example 3

hard
Find \lim_{x \to 0} \frac{\sin x}{x}

Common Mistakes

  • Assuming \lim_{x \to a} f(x) = f(a) always holds — the limit depends on nearby values, not the value at a itself, which may be undefined or different.
  • Confusing one-sided limits with the two-sided limit: \lim_{x \to a} f(x) exists only if both \lim_{x \to a^-} f(x) and \lim_{x \to a^+} f(x) exist and are equal.
  • Plugging in the value directly when the expression is indeterminate: \lim_{x \to 0} \frac{\sin x}{x} is not \frac{0}{0} — it requires algebraic or geometric reasoning to evaluate as 1.

Why This Formula Matters

Foundation of calculus—derivatives and integrals are defined using limits.

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?

\lim_{x \to a} f(x) = L

Why is the Limit formula important in Math?

Foundation of calculus—derivatives and integrals are defined using limits.

What do students get wrong about Limit?

The limit can exist even if f(a) doesn't. It's about approaching, not reaching.

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 →
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