Limits and continuity

The limit captures what value $f(x)$ approaches as $x$ gets arbitrarily close to a point $a$.

$$\lim_{x \to a} f(x) = L \;\;\Longleftrightarrow\;\; \forall\,\varepsilon>0,\;\exists\,\delta>0 :\; 0<|x-a|<\delta \implies |f(x)-L|<\varepsilon$$

Types of limits

• Finite limit at a point: $\lim_{x\to 3}\,(2x+1) = 7$.
• Infinite limit: $\lim_{x\to 0^{+}} \frac{1}{x} = +\infty$ (vertical asymptote).
• Limit at infinity: $\lim_{x\to\infty} \frac{1}{x} = 0$ (horizontal asymptote).

Indeterminate forms like $\tfrac{0}{0}$ or $\tfrac{\infty}{\infty}$ require algebraic manipulation (factoring, rationalising) or L'Hôpital's rule before the limit can be evaluated.

Continuity — $f$ is continuous at $a$ when three conditions hold simultaneously:

1. $f(a)$ exists.
2. $\lim_{x\to a} f(x)$ exists.
3. $\lim_{x\to a} f(x) = f(a)$.

A function continuous on a closed interval $[a,b]$ is guaranteed to attain every value between $f(a)$ and $f(b)$ (Intermediate Value Theorem).

Common pitfall: $\lim_{x\to a} f(x)$ is about the journey, not the destination: the limit can exist where $f(a)$ is undefined, and can differ from $f(a)$ where it is defined. Continuity is exactly the promise that they agree.