Limits: The Foundation of Calculus

Understand limits in calculus: what they mean, how to evaluate them algebraically and graphically, and worked examples for every common case.

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The core idea: getting close counts, even if you never arrive

A limit describes the value a function approaches as its input gets closer and closer to some number — even if the function never actually reaches that value at that exact point. It's the idea that makes calculus possible: before you can talk about instantaneous rate of change (derivatives) or the area under a curve (integrals), you need a rigorous way to talk about "approaching" a value. Limits are that tool.

The notation looks like this:
lim(x→a) f(x) = L

which reads: "as x approaches a, f(x) approaches L."

A limit only cares about what happens near a point, not necessarily at it. A function can have a limit at a point even if it's undefined there — that distinction is exactly why limits matter, and it's the single most common source of confusion for students first learning the topic.

Worked Examples

Example 1: Direct substitution

Easy

lim(x→3) (2x + 1)

Step 1: Plug in the value
2(3) + 1 = 7
Most limits of continuous functions can be solved by just plugging in the value.
Final Answer:
7

Example 2: A limit where direct substitution fails

Medium

lim(x→2) (x² - 4)/(x - 2)

Step 1: Check for direct substitution
Plugging in x = 2 gives 0/0 — undefined.
This is not "no limit exists", you must factor first.
Step 2: Factor and simplify
(x² - 4)/(x - 2) = (x - 2)(x + 2)/(x - 2) = x + 2
Cancel the common terms.
Step 3: Substitute the value
2 + 2 = 4
The limit is 4, even though the original function is undefined at x = 2.
Final Answer:
4

One-sided limits

For a piecewise function where the left and right sides don't match at a point, the limit only exists if both one-sided limits agree:

lim(x→a⁻) f(x) = lim(x→a⁺) f(x)

If they don't agree, the two-sided limit at that point does not exist (DNE) — even if the function is defined there.

Common mistakes to avoid

  • Assuming 0/0 means "no limit." It usually means you need to simplify first (factor, rationalize, or cancel) — 0/0 is a sign to keep working, not a final answer.
  • Confusing a limit with the function's actual value. lim(x→a) f(x) and f(a) can be different numbers, or one can exist while the other doesn't.
  • Forgetting to check both sides. A limit only exists if the function approaches the same value from the left and the right.

Where this shows up next

Limits are the definition behind the derivative. Once limits feel solid, derivatives are the natural next step.

Back to Calculus overview

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Derivatives

The derivative of a function at a point is literally defined as a limit of a slope as two points get infinitely close together.

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