Class 12 Statistics Notes · GSEB

Meaning of x Tends to a

Meaning of x Tends to a — understand x → a as an approach that never equals a, with left-hand (a⁻) and right-hand (a⁺) directions. GSEB Class 12 Commerce Statistics notes with approach diagrams.

Last updated: 23 Sep 2026

Notes

What Does x → a Mean?

x Tends to a (x → a)

x approaches arbitrarily close to a while never becoming equal to a — the values of x keep getting closer, the distance |x − a| shrinks toward zero but never reaches it.

Formal Meaning

x \to a \iff 0 < |x - a| < \delta \text{ for shrinking } \delta

Two ways to approach a — pick a side

x → a⁻ (left-hand)

Values of x less than a, sliding up toward a. Also written x → a − 0.

x → a⁺ (right-hand)

Values of x greater than a, sliding down toward a. Also written x → a + 0.

x → a never means x = a

“Tends to” is about getting close, not arriving. “Let x = a” is substitution — a different operation from taking a limit.

Try It: Walk Toward a = 2

Step through a sequence of values approaching 2 from either side. Watch the distance shrink — and confirm x never lands on 2.

Walking toward a = 2 — press Next to take another step

Step 1 of 5

x = 1.5

Distance from 2 = 0.5000 · x < 2, so x ≠ 2 always

1.51.91.991.9991.9999

Notice the pattern

The values squeeze toward 2 but never equal 2. Distance keeps shrinking — 0.5 → 0.1 → 0.01 → 0.001 → … — yet stays strictly positive. That is exactly what “x tends to a” means.

Left-Hand and Right-Hand Approaches

Every approach to a happens from exactly one side at a time. Naming the two directions explicitly lets us later check whether a limit exists — it does only when both sides agree.

Left-hand approach

x → a⁻ (x < a, x near a)

Start left of a and creep rightward: a − 3, a − 1, a − 0.1, a − 0.01, …
Every value satisfies x < a
Produces the left-hand limit (LHL)

Example: for a = 5: 4, 4.9, 4.99, 4.999 → distance 1 → 0.1 → 0.01 → 0.001

Right-hand approach

x → a⁺ (x > a, x near a)

Start right of a and creep leftward: a + 3, a + 1, a + 0.1, a + 0.01, …
Every value satisfies x > a
Produces the right-hand limit (RHL)

Example: for a = 5: 6, 5.1, 5.01, 5.001 → distance 1 → 0.1 → 0.01 → 0.001

The golden condition

The limit lim_(x→a) f(x) exists if and only if the left-hand approach and the right-hand approach arrive at the same value:

Limit Exists ⟺ LHL = RHL

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

Real life: agreeing on a meeting spot

Two friends walk toward a café from opposite streets. Same café → they meet (limit exists). Different cafés → no meeting (no limit). LHL and RHL must land on the same height.

Solved Examples

Solved Example

Problem

List four values that show x → 3 from the left, and four that show x → 3 from the right.

Solution

Left: 2, 2.5, 2.9, 2.99 · Right: 4, 3.5, 3.1, 3.01

Solved Example

Problem

Which of these statements are correct? (i) x → 5 means x = 5. (ii) x → 5⁻ uses values less than 5. (iii) |x − 5| → 0 as x → 5.

Solution

(ii) and (iii) are correct; (i) is wrong

Solved Example

Problem

A bike travels toward kilometre marker 12. Its positions are 10.5, 11.5, 11.9, 11.99 km. Which direction is it approaching from, and what is a?

Solution

Approaching from the left: x → 12⁻, a = 12

Key Takeaways

x → a means x gets close to a without ever equalling it — the journey, not the destination.
x → a⁻ approaches from the left (below a, gives LHL); x → a⁺ approaches from the right (above a, gives RHL).
A two-sided limit exists only when both sides converge to the same value — LHL = RHL.
Notation trap: the superscript − / + names the SIDE of approach, not the sign of x — x → 5⁻ uses 4.9, not −5.