a Limit?

The value a function approaches as its input gets arbitrarily close to a point — the very foundation that makes calculus possible.

Simple Definition

Imagine walking toward a wall. You get closer and closer, halving the distance each step — 1 meter, 0.5 m, 0.25 m, 0.125 m... You're approaching the wall, even if you never actually touch it. That's the idea of a limit.

In mathematics, the limit of a function $f(x)$ as $x$ approaches some value $a$ is the value that $f(x)$ gets closer and closer to as $x$ gets closer and closer to $a$. Crucially, we don't care what happens at $x = a$ — only what the function approaches.

This matters enormously because many functions are undefined at certain points (like $\frac{0}{0}$), yet they approach a perfectly clear value. Limits let us work with those cases precisely.

Intuition

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Approaching from both sides

A limit exists only if the function approaches the same value from both the left and the right. If you walk toward a cliff from both directions and the height you're stepping toward is different depending on which side you come from — there's no single limit. If both sides agree, the limit exists.

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The function's value doesn't matter

The limit of $f(x)$ as $x \to a$ is completely independent of $f(a)$. The function could be undefined at $a$, it could have a hole in its graph, or its value could be something totally different — and the limit would still exist and have its own value. This is what makes limits so powerful.

Visual — Approaching a Point

The function has a hole at x = 2, yet the limit clearly exists — both sides approach the same value

Numerical Intuition

Consider $\lim_{x \to 2} \dfrac{x^2 - 4}{x - 2}$. At $x=2$ this is $\frac{0}{0}$ — undefined. But watch what happens as we approach:

x (from left)	f(x)	x (from right)	f(x)
1.9	3.9	2.1	4.1
1.99	3.99	2.01	4.01
1.999	3.999	2.001	4.001
→ 2	→ 4	← 2	← 4

Both sides converge to 4. So $\lim_{x \to 2} \dfrac{x^2 - 4}{x - 2} = 4$, even though $f(2)$ is undefined.

Formal Definition

We write $\lim_{x \to a} f(x) = L$, and read it as "the limit of $f(x)$ as $x$ approaches $a$ equals $L$." The precise (epsilon-delta) definition:

Epsilon-Delta Definition

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

In plain English: no matter how small a window $\varepsilon$ you draw around $L$, you can always find a window $\delta$ around $a$ so that all $x$ inside that window map to $f(x)$ inside your $\varepsilon$-window. The function is truly, precisely approaching $L$.

Key Limit Laws

Sum	$\lim[f+g] = \lim f + \lim g$
Product	$\lim[f \cdot g] = \lim f \cdot \lim g$
Quotient	$\lim\dfrac{f}{g} = \dfrac{\lim f}{\lim g}\quad (\lim g \neq 0)$
Constant	$\lim[c \cdot f] = c \cdot \lim f$
Power	$\lim[f^n] = (\lim f)^n$
Squeeze Thm	If $g \leq f \leq h$ and $\lim g = \lim h = L$, then $\lim f = L$

Quick Examples

Example 1 — Direct Substitution

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

Plug in directly: $3^2 + 1 = 10$

Example 2 — Factor & Cancel

$\lim_{x \to 2} \dfrac{x^2-4}{x-2}$

Factor: $\dfrac{(x-2)(x+2)}{x-2} = x+2$
Result: $4$

Example 3 — Infinity

$\lim_{x \to \infty} \dfrac{1}{x}$

As $x$ grows without bound, $\frac{1}{x} \to 0$

Example 4 — Famous Limit

$\lim_{x \to 0} \dfrac{\sin x}{x}$

This equals $1$ — proven with the Squeeze Theorem, and essential in all of calculus.

Common Mistakes

✗ Confusing the limit with the function's value

The limit as $x \to a$ has nothing to do with $f(a)$. A function can have a limit of 5 at a point where it equals 0, or is completely undefined.

✗ Assuming a limit always exists

If the left-hand limit and right-hand limit are different, the (two-sided) limit does not exist. Always check both sides for piecewise or discontinuous functions.

✗ Writing $\frac{0}{0}$ = 0 or undefined without investigating

$\frac{0}{0}$ is an indeterminate form — it tells you nothing on its own. You must simplify, factor, or apply L'Hôpital's Rule to find the actual limit.

Related Concepts

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Sum	\(\lim[f+g] = \lim f + \lim g\)
Product	\(\lim[f \cdot g] = \lim f \cdot \lim g\)
Quotient	\(\lim\dfrac{f}{g} = \dfrac{\lim f}{\lim g}\quad (\lim g \neq 0)\)
Constant	\(\lim[c \cdot f] = c \cdot \lim f\)
Power	\(\lim[f^n] = (\lim f)^n\)
Squeeze Thm	If \(g \leq f \leq h\) and \(\lim g = \lim h = L\), then \(\lim f = L\)