The Concept of a Limit

By the end of this lesson, you will be able to estimate limits numerically and graphically, understand the formal definition intuitively, distinguish between the value of a function at a point and the limit at that point, and identify when limits don't exist.

The Big Idea

The limit is the central concept of calculus. It answers: "what value does $f(x)$ approach as $x$ gets arbitrarily close to $c$?" — NOT what $f(c)$ equals. The distinction matters enormously: the function might not even be defined at $c$ (like $\frac{\sin x}{x}$ at $x = 0$), yet the limit can still exist.

Limits enable us to rigorously define derivatives, integrals, and continuity — the foundations of all calculus.

Intuitive Definition

$$\lim_{x \to c} f(x) = L$$

means: "$f(x)$ can be made as close to $L$ as desired by taking $x$ sufficiently close to $c$ (but not equal to $c$)."

Key points:

• $x$ approaches $c$ from both sides
• $f(c)$ need not be defined — only the approach matters
• The limit is unique if it exists

Estimating Limits Numerically

Build a table: choose $x$-values approaching $c$ from both left ($x \to c^-$) and right ($x \to c^+$).

Example 1 — Numerical Estimation

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

$x$	$f(x)$
1.9	3.9
1.99	3.99
1.999	3.999
2.001	4.001
2.01	4.01
2.1	4.1

The limit appears to be 4. (Note: $f(2)$ is undefined — $0/0$ — but the limit exists.)

Algebraically: $\dfrac{(x-2)(x+2)}{x-2} = x+2$ for $x \neq 2$; as $x \to 2$, this $\to 4$.

Graphical Interpretation

The limit is the $y$-value that the graph approaches (from both sides) as $x \to c$.

• If the graph approaches the same value from both sides → limit exists
• If the graph approaches different values → limit doesn't exist (jump)
• If the graph goes to $\pm\infty$ → limit doesn't exist (vertical asymptote)