Limits and continuity

A limit describes what a function approaches, not necessarily what it equals. We write

$$\underset{x\to a}{\lim }f(x)=L$$

when $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $a$.

Limits are the foundation under both Derivative and Integral. A derivative is a limit of secant slopes, while a definite integral is a limit of Riemann sums. Without limits, calculus is just a list of rules; with limits, the rules have meaning.

Continuity

A function is continuous at $a$ when three things agree:

$$\underset{x\to a}{\lim }f(x)=f(a).$$

Plain-language version: no jump, hole, or blow-up at that point. Continuity on an interval is often enough for the Fundamental theorem of calculus to work in its standard first-year form.

One-sided limits

Sometimes left and right approaches differ:

$$\underset{x\to a^{-}}{\lim }f(x),\underset{x\to a^{+}}{\lim }f(x).$$

The two-sided limit exists only if both one-sided limits exist and are equal.

Postgraduate bridge

In real analysis, limits are formalised with $\epsilon$-$\delta$ definitions. In topology, continuity means inverse images of open sets are open. These are the same idea at different abstraction levels: nearby inputs must map to nearby outputs.

Common pitfalls

• Substituting too early when the expression gives $0/0$; simplify or use a theorem first.
• Assuming a function must be defined at $a$ for its limit as $x\to a$ to exist.
• Forgetting to check both sides at jumps, absolute values, and piecewise definitions.