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What is a Limit?

A limit describes the value a function approaches as its input gets closer to a particular point. Limits are the foundation of calculus, underlying both derivatives and integrals.

Short answer

The limit of f(x) as x approaches c is the value f(x) gets arbitrarily close to, written lim(x→c) f(x) = L. For continuous functions, you can often find it by direct substitution.

f(x) approaching a limit as x→c

x: x · y: f(x)

Try it: interactive calculator

Limit value L

= 2*3+1

Slope m2 Point x approaches (c)3 Intercept b1

Step-by-step worked examples

Find lim(x→2) (3x+4).

Since 3x+4 is continuous, substitute x=2 directly.
3(2)+4 = 6+4 = 10

Find lim(x→3) (x²−9)/(x−3).

Direct substitution gives 0/0 (indeterminate).
Factor: (x²−9)/(x−3) = (x−3)(x+3)/(x−3) = x+3
Substitute x=3: 3+3 = 6

Find lim(x→0) (sin x)/x.

This is a well-known limit that cannot be solved by substitution (0/0).
Using the standard trigonometric limit result: lim(x→0) sin(x)/x = 1

Flashcards

Quick quiz

Q1.lim(x→1) (2x+5) = ?

Correct answer: A. Substitute x=1: 2(1)+5=7.

Q2.lim(x→4) (x²−16)/(x−4) = ?

Correct answer: C. Factor to (x-4)(x+4)/(x-4)=x+4; at x=4 this gives 8.

Q3.What does an indeterminate form like 0/0 signal?

Correct answer: B. You must simplify before the limit can be evaluated.

Q4.lim(x→0) sin(x)/x = ?

Correct answer: B. This is a standard trigonometric limit equal to 1.

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Common mistakes

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Plugging in the value even when it gives 0/0 and stopping there. — Correct: 0/0 means you must simplify first (factor, rationalize) before evaluating.

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Thinking a limit only exists if f(c) is defined. — Correct: A limit can exist even if f(c) is undefined — it's about nearby behavior, not the exact point.

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Confusing lim(x→c) f(x) with f(c). — Correct: They can differ; the limit describes the approach, f(c) is the actual value there.

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Assuming all limits require substitution to solve. — Correct: Some limits (like sin x/x) need special techniques or known results.

FAQ

What is a limit in calculus?

A limit is the value a function approaches as its input gets arbitrarily close to a given point, written lim(x→c) f(x) = L.

What is the limit formula?

There's no single formula — limits are evaluated by direct substitution for continuous functions, or by factoring/simplifying for indeterminate forms like 0/0.

What are examples of limits?

lim(x→2)(3x+4)=10 by substitution, and lim(x→3)(x²−9)/(x−3)=6 after factoring out the common term.

How to calculate a limit?

Try direct substitution first; if you get an indeterminate form like 0/0, factor, simplify, or use a known standard limit result.