Limit — Mathematics Knowledge System

Overview

The limit describes the value a function approaches as its input approaches some point, without necessarily ever reaching it. Limits are the rigorous foundation underneath every other idea in calculus: the derivative is defined as a limit, and the integral is defined as a limit of sums.

Intuition

Imagine walking toward a wall, each step covering half the remaining distance. You never technically touch the wall, but you get arbitrarily close — closer than any distance someone could name. That's a limit: it's not about arriving, it's about the fact that you can get as close as anyone demands, just by taking enough steps. f(x) → L as x → a means: no matter how tiny a target distance you pick around L, f(x) eventually lands (and stays) inside that target as x gets close enough to a.

Formal Definition

Definition

The epsilon-delta definition (Weierstrass):

\lim_{x \to a} f(x) = L \iff \forall \varepsilon > 0,\ \exists \delta > 0 \text{ such that } 0 < |x-a| < \delta \Rightarrow |f(x)-L| < \varepsilon

For every tolerance ε around L, there is a corresponding neighborhood δ around a within which f(x) stays inside that tolerance.

Notation

Notation	Meaning
$\lim_{x \to a} f(x)$	The limit of f(x) as x approaches a
$\lim_{x \to a^+} f(x)$	Right-hand limit — x approaches a from values greater than a
$\lim_{x \to a^-} f(x)$	Left-hand limit — x approaches a from values less than a
$\lim_{x \to \infty} f(x)$	The value f(x) approaches as x grows without bound

Derivation

A limit exists at a point if and only if the left-hand and right-hand limits both exist and are equal:

\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L

Two-sided limit exists exactly when both one-sided limits agree

Properties

Sum rule

\lim_{x\to a}[f(x)+g(x)] = \lim_{x\to a}f(x) + \lim_{x\to a}g(x)

Product rule

\lim_{x\to a}[f(x)g(x)] = \lim_{x\to a}f(x) \cdot \lim_{x\to a}g(x)

Quotient rule

\lim_{x\to a}\frac{f(x)}{g(x)} = \frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}

Condition: provided the denominator limit is nonzero

Squeeze theorem

\text{If } g(x)\leq f(x) \leq h(x) \text{ near } a \text{ and } \lim g = \lim h = L, \text{ then } \lim f = L

Example: Used to prove \lim_{x\to 0}\frac{\sin x}{x}=1

Applications

Instantaneous velocity is defined as the limit of average velocity as the time interval shrinks to zero.

Formula Explorer

Worked Examples

This function is continuous everywhere, so the limit equals the function value at x=3.
$2(3)+1 = 7$

Answer: 7

Practice Problems

Difficulty 3/10

Evaluate lim(x→0) sin(x)/x.

Difficulty 2/10

What is lim(x→2) (x² - 4)/(x - 2)?

024Undefined

Common Mistakes

Common Mistake

Believing a limit doesn't exist just because the function is undefined at that exact point.

Limits describe behavior NEAR a point, not AT it. (x²-1)/(x-1) is undefined at x=1 but its limit there is 2.

Common Mistake

Treating 0/0 as automatically equal to 0 or undefined without further work.

0/0 is an 'indeterminate form' — it signals you must simplify (factor, rationalize, or apply L'Hôpital's rule) to find the actual limit.

Quiz

What does lim(x→a) f(x) = L formally guarantee?

f(a) = Lf is continuous at aFor every ε>0 there is a δ>0 such that f(x) stays within ε of L whenever x is within δ of a (but not equal to a)f is differentiable at a

Flashcards

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Historical Background

Newton and Leibniz built calculus in the 1660s–1680s using 'infinitesimals' — infinitely small quantities — which worked but lacked logical rigor and drew sharp criticism (Bishop Berkeley famously called them 'ghosts of departed quantities' in 1734). It took over a century to fix this. Augustin-Louis Cauchy gave the first reasonably rigorous treatment of limits in his 1821 Cours d'Analyse, and Karl Weierstrass finalized the modern epsilon-delta definition in the 1850s–60s, finally putting calculus on solid logical footing.

1660s–1680s
Newton and Leibniz develop calculus using infinitesimals
Isaac Newton, Gottfried Wilhelm Leibniz
1734
Berkeley publishes 'The Analyst', attacking the lack of rigor in infinitesimals
George Berkeley
1821
Cauchy formalizes limits in Cours d'Analyse
Augustin-Louis Cauchy
1861
Weierstrass gives the modern epsilon-delta definition
Karl Weierstrass

Summary

A limit describes the value f(x) approaches as x approaches a, without requiring f(a) to be defined.
Formal definition (Weierstrass): ∀ε>0 ∃δ>0 such that 0<|x-a|<δ ⟹ |f(x)-L|<ε.
A two-sided limit exists only if the left and right limits agree.
Indeterminate forms (0/0, ∞/∞) require algebraic manipulation — factoring, rationalizing, L'Hôpital's rule.
Limits are the rigorous foundation for both the derivative and the integral.

References

BookCauchy, A.-L. (1821). Cours d'Analyse de l'École Royale Polytechnique.
WebsiteWikipedia — Limit of a function