A limit describes the value a function approaches as its input creeps towards some number $a$ — not the value at $a$, which the function may never actually reach (or may not even be defined there). We write $\lim_{x \to a} f(x) = L$ to mean: as $x$ gets arbitrarily close to $a$, the output $f(x)$ gets arbitrarily close to $L$.
One-sided limits. Because $x$ can approach $a$ from two directions, we distinguish the left-hand limit (LHL), approaching through values smaller than $a$, and the right-hand limit (RHL), approaching through larger values:
Existence of a limit. The limit at $a$ exists if and only if both one-sided limits exist and agree. This is the single most-tested idea of the topic:
If the two sides disagree — as happens at a jump — the limit simply does not exist, no matter how nicely the function behaves elsewhere.
Algebra of limits. Provided $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ both exist, limits respect the basic operations. Let $\lim f = L$ and $\lim g = M$:
| Law | Statement |
|---|---|
| Sum / Difference | $\lim (f \pm g) = L \pm M$ |
| Constant multiple | $\lim (k\,f) = kL$ |
| Product | $\lim (f \cdot g) = L \cdot M$ |
| Quotient | $\lim \dfrac{f}{g} = \dfrac{L}{M}, \;\; M \ne 0$ |
| Power | $\lim \big(f\big)^{n} = L^{n}$ |
Polynomials and rational functions. Polynomials are continuous everywhere, so for any polynomial $p(x)$ you get the limit by direct substitution: $\lim_{x \to a} p(x) = p(a)$. For a rational function $\dfrac{p(x)}{q(x)}$, substitution works as long as $q(a) \ne 0$.
The $\dfrac{0}{0}$ form. When both numerator and denominator vanish at $a$, substitution gives the meaningless symbol $\dfrac{0}{0}$ — an indeterminate form. The cure is to factorise and cancel the common factor $(x - a)$ that is causing both to vanish, then substitute into what remains:
Deeper Insight — a limit is about the journey, not the destination: The whole power of the limit concept comes from a deliberate separation between what a function does near a point and what it is at that point. In the example above, $\dfrac{x^2-4}{x-2}$ is genuinely undefined at $x = 2$ — there is a hole in the graph — yet the limit is a perfectly definite $4$, because every nearby value of the function is close to $4$. This is exactly why cancelling the factor $(x-2)$ is legitimate: for all $x \ne 2$ the two expressions are identical, and the limit only ever cares about $x \ne a$. The $\dfrac{0}{0}$ symbol is not an answer but a signal that a hidden common factor is masking the true behaviour, and learning to read that signal — factor, rationalise, or simplify — is the skill that the entire calculus syllabus is built upon.