Intuitive Idea and Algebra of Limits

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 $L$. A quick numerical table makes the idea concrete — for $f(x) = \dfrac{x^2 - 1}{x - 1}$ near $x = 1$, feeding in $0.9, 0.99, 0.999$ and $1.1, 1.01, 1.001$ produces outputs marching towards $2$ from both sides, even though $f(1)$ itself is undefined.

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:

To compute a one-sided limit of a piecewise function, you must use the piece that applies on that side of $a$ — the rule for $x < a$ for the LHL, the rule for $x > a$ for the RHL.

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. A classic case is $f(x) = |x|/x$ at $x = 0$: the LHL is $-1$ and the RHL is $+1$, so $\lim_{x \to 0} f(x)$ does not exist.

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 $\lim g = M$:

These laws are why limits are computable at all: a complicated expression is broken into pieces, each piece's limit is taken separately, and the results are recombined. The one trap is the quotient law — it is only valid when the denominator's limit $M \ne 0$.

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 $q(a) \ne 0$, in which case $\lim_{x \to a} \dfrac{p(x)}{q(x)} = \dfrac{p(a)}{q(a)}$.

The $\dfrac{0}{0}$ form. When both numerator and denominator vanish at $a$, substitution gives the meaningless symbol $\dfrac{0}{0}$ — an indeterminate form. It does not mean the limit fails to exist; it means more work is needed. There are two standard cures:

• Factor and cancel: when both parts are polynomials, $(x - a)$ is a common factor of each. Factor it out, cancel it, then substitute.
• Rationalise: when a square root is present, multiply numerator and denominator by the conjugate to convert the surd difference into a difference of squares, which then cancels the offending factor.

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.