Limits and continuity

Understanding Limits

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

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

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

Continuity and Pitfalls

A function is continuous at $a$ when three conditions hold simultaneously:

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

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

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 defined. Continuity is the exact promise that they agree.