Existence of a limit

Push a value of $x$ closer and closer to some fixed point $a$ — never landing on it, just creeping toward it — and watch what $f(x)$ does. If the outputs settle down and crowd around one single number no matter which side you approach from, that number is the limit of $f(x)$ at $a$ . If the outputs disagree depending on the side, or refuse to settle at all, the limit simply does not exist. This question of existence is the very first idea Class 11 calculus asks you to master, before any formula for computing a limit's value is introduced. It matters because differentiation itself is built entirely on limits, so a shaky grip on "does this limit exist" becomes a shaky grip on everything that follows. This article sharpens the left-hand/right-hand test into a precise two-condition rule, walks through a fully worked piecewise example, and flags the traps that trip up almost every first attempt.

The core insight

A limit $\lim_{x\to a} f(x)$ exists exactly when the left-hand approach and the right-hand approach converge to the same finite number, and that shared number is completely independent of whatever $f(a)$ happens to be.

The core idea

A limit is a statement about trend, not about a single output. Writing $\lim_{x\to a} f(x) = L$ means: as $x$ is chosen nearer and nearer to $a$ (from either side, but never equal to $a$ ), the values $f(x)$ crowd nearer and nearer to the fixed number $L$ . Two separate approach-paths lead into $a$ on the number line — from below and from above — and both are given their own names.

\lim_{x\to a^-} f(x) = L_1 \qquad \lim_{x\to a^+} f(x) = L_2

where: $\lim_{x\to a^-} f(x)$ is the left-hand limit (LHL), the trend of $f(x)$ as $x$ climbs up toward $a$ through values less than $a$ ; $\lim_{x\to a^+} f(x)$ is the right-hand limit (RHL), the trend as $x$ descends toward $a$ through values greater than $a$ . The existence rule glues these two together:

\lim_{x\to a} f(x) \text{ exists} \iff L_1 = L_2 = L

where: $L$ is a single finite number that both one-sided trends must land on. If $L_1 \ne L_2$ , or either side fails to settle to any fixed number (it blows up or oscillates), the two-sided limit does not exist — full stop, regardless of what $f(a)$ is. Once existence is confirmed, limits of combined expressions can be built from limits of simpler pieces, using rules that the ISC syllabus states without proof:

\lim_{x\to a}[f(x)\pm g(x)] = \lim_{x\to a}f(x) \pm \lim_{x\to a}g(x), \qquad \lim_{x\to a}\frac{f(x)}{g(x)} = \frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)} \ \ \big(\lim_{x\to a}g(x)\neq 0\big)

where: $f(x)$ and $g(x)$ are any two functions whose individual limits at $a$ already exist, and the quotient rule additionally needs the denominator's limit to be nonzero.

Figure

Two number lines showing x approaching a from the left giving output trend L1 and from the right giving output trend L2, with a check mark when they match and a cross when they differ

Derivation AThe LHL–RHL existence criterion

The formal criterion behind every existence check in this chapter is that a two-sided limit exists precisely when both one-sided limits exist and are equal, and the goal of this reasoning is to turn a vague picture of "the graph approaching a value" into an operational test that can be applied to any piecewise or algebraic function without ever drawing the graph.

Step 1. Isolate the one-sided behaviour. For $x \to a^-$ , only values of $x$ strictly less than $a$ are relevant, so if $f$ is defined piecewise, use whichever piece applies for $x < a$ . Symmetrically, for $x \to a^+$ , use whichever piece applies for $x > a$ .

Step 2. Compute (or numerically estimate via a table of values creeping toward $a$ ) the trend $L_1 = \lim_{x\to a^-} f(x)$ using only that left-side piece, and independently the trend $L_2 = \lim_{x\to a^+} f(x)$ using only the right-side piece.

Step 3. Compare $L_1$ and $L_2$ . If they are the same finite number, that shared value is declared the limit, $\lim_{x\to a} f(x) = L_1 = L_2$ , and it exists. If they differ, or if either trend fails to settle to a finite number, the two-sided limit does not exist, and no further computation can rescue it.

Step 4. Note explicitly what was never used: the value $f(a)$ . The criterion never asks what happens exactly at $a$ — a limit can exist at a point where $f(a)$ is undefined (a hole in the graph), or where $f(a)$ is defined but disagrees with $L$ (a redefined point); both are irrelevant to existence.

Worked example

Examine whether $\lim_{x\to 0} f(x)$ exists, where

f(x) = \begin{cases} x+1, & x < 0 \\ x-1, & x \ge 0 \end{cases}

Step 1 — set up the left-hand approach. For $x < 0$ , use $f(x) = x + 1$ . Take $x = -0.1, -0.01, -0.001,\dots$ , giving $f(-0.1) = 0.9$ , $f(-0.01) = 0.99$ , $f(-0.001) = 0.999$ . These values climb toward $1$ , so $L_1 = \lim_{x\to 0^-} f(x) = 1$ .

Step 2 — set up the right-hand approach. For $x > 0$ , use $f(x) = x - 1$ . Take $x = 0.1, 0.01, 0.001,\dots$ , giving $f(0.1) = -0.9$ , $f(0.01) = -0.99$ , $f(0.001) = -0.999$ . These values sink toward $-1$ , so $L_2 = \lim_{x\to 0^+} f(x) = -1$ .

Step 3 — compare. $L_1 = 1$ and $L_2 = -1$ are different numbers, so the LHL–RHL criterion fails immediately.

$\lim_{x\to 0} f(x)$ does not exist.

Competitive stretch

For the polynomial $g(x) = x^2 - 1$ , check $\lim_{x\to 2} g(x)$ . There is no piecewise split needed since $g$ is a single smooth expression, so both one-sided trends are computed from the same formula: $g(1.99) \approx 2.9601$ and $g(2.01) \approx 3.0401$ , both closing in on $3$ . Here $L_1 = L_2 = 3$ , so the limit exists and equals $3$ — and, only incidentally, this also matches $g(2) = 3$ , since polynomials happen to have no gaps or jumps anywhere.

Common traps

Assuming $\lim_{x\to a} f(x) = f(a)$ always holds — existence depends only on the two one-sided trends, and $f(a)$ can be undefined, different, or irrelevant.
Checking only one side of $a$ and declaring the limit exists — both $L_1$ and $L_2$ must be found and compared before any conclusion is valid.
Mixing up "limit does not exist" with "limit equals zero" or "function is undefined at $a$ " — these are three distinct statements about different things.
Forgetting that a piecewise function must use the correct branch for each side — plugging the $x \ge 0$ formula into an $x \to 0^-$ calculation gives a wrong $L_1$ .

Ask the tutor: ask your AI tutor to build a fresh piecewise or rational function and walk through the LHL–RHL comparison with you step by step.

Does a limit failing to exist mean the function is undefined at that point?

No — these are separate facts. A function can be perfectly defined at $a$ (say $f(a) = 10$ ) while its limit at $a$ still fails to exist, if the left and right trends disagree or blow up. Conversely, a function can be undefined at $a$ (a hole) while its limit exists just fine, since the limit only cares about points near $a$ , never at $a$ itself.

Why do we even need both a left-hand and a right-hand limit — isn't approaching from one side enough?

Approaching from only one side tells you only half the story. A function like a step function or an absolute-value-style piecewise rule can behave completely differently just before and just after $a$ . The two-sided limit is meant to capture the function's single, unambiguous trend at $a$ , so both directions must be checked and must agree before that single trend can be declared to exist.

If a limit does not exist, can we still say anything useful about the function there?

Yes. Reporting $L_1$ and $L_2$ separately is still meaningful and often required — for instance, stating "the left-hand limit is 1 and the right-hand limit is negative 1" describes a genuine jump in the graph. "Does not exist" for the two-sided limit does not mean no information is available; it means no single shared number describes both sides.

Existence of a limit

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads