Limits

A limit answers the question: as $x$ gets closer and closer to some number, what does $f (x)$ get closer and closer to? Notice the wording: closer and closer, not equal to. The limit is about the neighbourhood of a point, not the point itself.

Approaching a value

Consider $f (x) = \frac{x^{2} - 1}{x - 1}$ . At $x = 1$ the formula is $0 / 0$ , undefined. But for every other $x$ we can cancel: $\frac{(x - 1) (x + 1)}{x - 1} = x + 1$ . As $x$ slides toward $1$ , $f (x)$ slides toward $2$ , even though $f (1)$ itself does not exist. We write

lim_{x \to 1} \frac{x^{2} - 1}{x - 1} = 2.

The hole in the graph at $x = 1$ does not change where the function is heading.

One-sided limits

Sometimes the answer depends on which way you approach. For $g (x) = \frac{| x |}{x}$ , coming in from the right ( $x > 0$ ) gives $+ 1$ ; from the left ( $x < 0$ ) gives $- 1$ . The two one-sided limits disagree, so the two-sided limit does not exist. A limit exists only when both sides agree.

How close is close enough

Informally, $lim_{x \to a} f (x) = L$ means we can force $f (x)$ to be as close to $L$ as we like (within any tolerance we name) just by keeping $x$ close enough to $a$ . Making that "as close as we like" precise is the famous epsilon-delta definition, but the intuition is what you will use daily.

Continuity

A function is continuous at $a$ when nothing surprising happens there: the limit exists, $f (a)$ exists, and the two match,

lim_{x \to a} f (x) = f (a) .

Graphically, you can draw it through that point without lifting your pen: no holes, no jumps. Continuous functions are the well-behaved ones on which the rest of calculus is built.