Class 12 Statistics Notes · GSEB

Limit of a Function

Limit of a Function — master lim(x→a) f(x) = L, left-hand and right-hand limits, the tabular method, and the rule that a limit exists only when LHL equals RHL. GSEB Class 12 Commerce Statistics notes with solved examples.

Last updated: 23 Sep 2026

Notes

Limit of a Function

We write lim(x→a) f(x) = L when the values of f(x) approach the single number L as x approaches a from both sides — regardless of whether f(a) is defined, and regardless of what f(a) equals if it is defined.

Limit Exists ⟺ LHL = RHL

\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L

One-Sided Limits via h

\text{LHL} = \lim_{h \to 0} f(a - h), \quad \text{RHL} = \lim_{h \to 0} f(a + h)

Your speedometer shows 42 km/h the instant you brake — that is a limit: average speed over smaller and smaller time windows, settling on one number. You never measure “over zero seconds”; you approach zero.

The Golden Rule: LHL Must Equal RHL

LHL = RHL = L

Both sides agree → limit exists and equals L

LHL ≠ RHL

Sides disagree → limit does not exist (jump)

LHL = RHL ≠ f(a)

Limit still exists — f(a) is irrelevant (removable hole)

f(a) never decides the limit

The limit looks near a, not at a. Change or delete f(a) — the limit does not move. The value f(a) is a separate question (continuity), asked later.

When Do We Use the Limit?

Three situations where you need a limit instead of plain substitution:

1. Substitution fails

Plugging x = a gives 0/0 — the expression breaks. The limit still has an answer.

2. f(a) does not exist

The function is undefined at the point — but values around it still settle on a number.

3. Base for what comes next

Continuity and differentiation (next chapters) are both built on this idea of approaching.

Try It: Tabular Method

Make a table of x approaching a from both sides. Whichever value f(x) settles on is the limit. Pick a function below and watch the two columns.

Tabular Method — values squeeze toward a from both sides

Left side — x → 3⁻

x	f(x)
2	5
2.5	5.5
2.9	5.9
2.99	5.99
2.999	5.999

Right side — x → 3⁺

x	f(x)
4	7
3.5	6.5
3.1	6.1
3.01	6.01
3.001	6.001

Both columns converge to

lim(x→3) f(x) = 6

LHL ≈ 6 · RHL ≈ 6 · LHL = RHL ✓ · f(3) undefined — yet the limit survives

Read the table like a story

f(3) is undefined (0/0), yet both sides settle on 6 — the limit exists even though the function does not.

Graphical Picture: The Hole

On a graph, the limit is the height both sides climb toward. If f(a) is undefined, that height has a punched-out dot — a hole — but the limit still reads that height.

The hole at x = a — limit cares about the approach, not the point

From left

f(x) → L

From right

f(x) → L

At x = a

f(a) can be anything (or nothing)

The hollow circle's height is the limit.

Solved Examples

Solved Example

Problem

Using a table of values, find lim(x→3) (x² − 9)/(x − 3).

Solution

lim(x→3) (x² − 9)/(x − 3) = 6

Solved Example

Problem

If LHL of f at 2 is 5 and RHL is 8, does lim(x→2) f(x) exist?

Solution

No — the limit does not exist (jump at x = 2)

Solved Example

Problem

f(x) = x² for x ≠ 4, and f(4) = 100. What is lim(x→4) f(x)?

Solution

lim(x→4) f(x) = 16 (f(4) = 100 is irrelevant)

Key Takeaways

lim(x→a) f(x) = L means f(x) approaches L as x approaches a from both sides — f(a) plays no role.
The limit exists if and only if LHL = RHL; a mismatch means no limit (jump).
Use limits when substitution gives 0/0, when f(a) does not exist, and as the base for continuity and differentiation.
Tabular method: values of f(x) squeezing toward a from left and right — whichever number both columns settle on is the limit.
A hole at (a, L) with matching sides still gives limit L — the limit reads the map around a, not the pin at a.