What a Limit Really Asks

A limit answers one question: as the input $x$ crowds in on some point $x_0$ from both sides, what value does $f(x)$ crowd in on? Crucially, this is a statement about the neighbourhood of $x_0$, not about the point itself. Whether $f$ is even defined at $x_0$ — or what value it takes there if it is — is irrelevant to whether the limit exists.

Definition 9.1 (informal). Let $I$ be an open interval containing $x_0$ and let $f: I \to \mathbb{R}$. We say $\lim_{x \to x_0} f(x) = L$ if, whenever $x$ is taken sufficiently close to $x_0$ (from either side, with $x \neq x_0$), $f(x)$ gets correspondingly close to $L$.

A worked illustration. Take $f(x) = x^2 + 1$ near $x = 3$. Plugging in values like $2.9, 2.99, 2.999$ and $3.1, 3.01, 3.001$ shows $f(x)$ creeping toward $10$ from both sides — and indeed $f(3) = 10$ too. Here the limit and the function value agree, which is the friendly case. But that agreement is a bonus, not a requirement of the definition.

One-sided limits

Because "approaching from either side" is doing real work in the definition, it helps to split it into two directions.

• Left-hand limit (Definition 9.2): $\lim_{x \to x_0^-} f(x) = l_1$ — the value $f(x)$ settles toward as $x$ increases up to $x_0$ while staying less than $x_0$.
• Right-hand limit (Definition 9.3): $\lim_{x \to x_0^+} f(x) = l_2$ — the value $f(x)$ settles toward as $x$ decreases down to $x_0$ while staying greater than $x_0$.

The reconciliation theorem: the two-sided limit $\lim_{x \to x_0} f(x)$ exists and equals $L$ if and only if both one-sided limits exist and agree:

If even one of the one-sided limits fails to exist, or the two one-sided limits disagree, the two-sided limit simply does not exist — there is no "in-between" value to fall back on.

Reading limits from tables and graphs

Before any algebraic machinery, limits can be estimated two ways:

1. Table of values — plug in $x$-values that approach $x_0$ from below and above, and watch what $f(x)$ trends toward.
2. Graph — trace the curve with your eye from both sides of $x_0$ and see where the two traces converge (or fail to).

When limits genuinely fail to exist

The classic culprit is a function that behaves differently on the two sides of the point.

Example. Consider $g(x) = \dfrac{x}{|x|}$ (undefined at $x=0$). For $x>0$, $\frac{x}{|x|}=1$; for $x<0$, $\frac{x}{|x|}=-1$. So

These disagree, so $\lim_{x\to 0} g(x)$ does not exist — even though $g$ is perfectly well-behaved (constant, even) on either side individually. This is the signature pattern of a "jump" in behaviour: two honest, finite one-sided limits that simply don't match.

Similarly, a function like the greatest-integer (floor) function has a limit that fails to exist at every integer, for exactly this reason: approaching an integer $n$ from below gives $n-1$, from above gives $n$.

The independence of $f(x_0)$ from the limit

This is the single most important conceptual point in the whole idea of a limit, and it cuts two ways:

1. The limit can exist even when $f(x_0)$ does not. For instance, $f(x) = \dfrac{x^2-16}{x+4}$ is undefined at $x=-4$ (division by zero), yet for every $x \neq -4$ it simplifies to $f(x) = x - 4$, so as $x \to -4$ from either side, $f(x) \to -8$. The graph has a "hole" at $x=-4$, but the limit exists perfectly well there.
2. The limit can exist and disagree with $f(x_0)$, or $f(x_0)$ can exist while the limit doesn't. The value of the function at the point plays no role whatsoever in computing $\lim_{x\to x_0} f(x)$ — that computation only ever looks at points near $x_0$, deliberately excluding $x_0$ itself.

Finally, note that "the limit is a real number $L$" is itself part of the claim — a limit is only said to exist when that number is unique and finite. If the one-sided approach values run off unboundedly (as happens with $1/x^2$ near $0$) or never settle down at all, we again say the limit does not exist, even though we may still describe that runaway behaviour using the symbol $\infty$ (a topic for later).