1. Motivation — Why Do We Need Limits?
The Central Question
What value does a function approach as its input gets close to a particular point — even if the function is not defined there, or even if its defined value is "wrong"?
1.1 A Function Undefined at the Point
Consider the function:
$$ f(x) = \frac{x^2 - 1}{x - 1} $$
At $x = 1$, the denominator is zero, so $f(1)$ is undefined. Yet for every $x \neq 1$ we can factor:
$$ f(x) = \frac{(x-1)(x+1)}{x-1} = x + 1 \quad (x \neq 1) $$
As $x$ gets closer and closer to $1$, $f(x)$ gets closer and closer to $2$. We write:
$$ \lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2 $$
The limit captures the trend of the function near the point, not its value at the point.
1.2 A Function Whose Value Disagrees with the Trend
Define a piecewise function:
$$ g(x) = \begin{cases} x + 1 & \text{if } x \neq 1 \\ 5 & \text{if } x = 1 \end{cases} $$
Here $g(1) = 5$, but for all nearby points $g(x) = x + 1 \to 2$. So $\lim_{x \to 1} g(x) = 2$, even though $g(1) = 5$. The limit and the function value are different!
1.3 Graphical Exploration
The interactive graph below shows $f(x) = \frac{x^2 - 1}{x - 1}$. Notice the hole at $(1, 2)$ — the function is not defined there, but the graph clearly approaches the point.
2. The Epsilon–Delta ($\varepsilon$–$\delta$) Definition
Let $f$ be defined on an open interval containing $a$, except possibly at $a$ itself. We say
$$ \lim_{x \to a} f(x) = L $$
if for every $\varepsilon > 0$ there exists a $\delta > 0$ such that