Limits and Continuity

Definition

The limit of a function $f(x)$ as $x$ approaches $a$ is the value that $f(x)$ approaches:

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

A function $f$ is continuous at $a$ if $f(a)$ is well-defined and finite, and

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

A function is continuous over an interval if its graph forms an unbroken curve — no holes, no jumps, no infinite divergence at any finite point.

Intuition

A limit captures the behaviour of a function near a point without requiring the function to be defined there. Continuity says the function has no surprises: zooming in on the graph always reveals a connected curve, and evaluating at the point gives the same answer as approaching from either side.

Formal Description

Direct substitution often yields an indeterminate form $\frac{0}{0}$. Algebraic manipulation can resolve this. For example:

$$\underset{h\to 0}{\lim }\frac{(x+h{)}^2-x^2}{h}=\underset{h\to 0}{\lim }\frac{h(2x+h)}{h}=2x.$$

Intermediate Value Theorem: If $f$ is continuous on $[a,b]$, then $f$ takes every value between $f(a)$ and $f(b)$ on that interval.

Applications

Limits underpin the definition of the derivative (${\lim }_{h\to 0}$) and the Riemann integral. Continuity is a prerequisite for many theorems in calculus, including the Intermediate Value Theorem and the Extreme Value Theorem.

Trade-offs

Limits can fail to exist if the left- and right-hand limits differ (jump discontinuity) or if the function oscillates infinitely near the point. Continuity at a point does not imply differentiability there (e.g. $|x|$ at $x=0$).

Links

• Newton’s Method — uses limits implicitly via derivatives