Class 12 Statistics Notes · GSEB

Standard Forms

Standard Forms — use direct substitution for polynomial limits and the key formula (xⁿ − aⁿ)/(x − a) → n·aⁿ⁻¹ to resolve 0/0 forms quickly. GSEB Class 12 Commerce Statistics notes with step-by-step factorisations.

Last updated: 23 Sep 2026

Notes

What Are Standard Forms?

Standard Forms of Limits

Ready-made results that turn common limit patterns into instant answers. Two matter here: polynomials (plug in directly) and (xⁿ − aⁿ)/(x − a) (answer n·aⁿ⁻¹).

Like a cook's memorised ratio (2 rice : 3 water) — recognised patterns answered from memory, so you spend working marks on the unusual parts only.

Standard Form 1 — Polynomials (Direct Substitution)

Just plug in

Polynomial Limit

\lim_{x \to a} P(x) = P(a)

Any expression built from +, −, × and whole-number powers is a polynomial — substitute without hesitation.

Examples (all direct substitution):

lim(x→2) (3x² − 5x + 1) = 12 − 10 + 1 = 3

lim(x→−1) (x³ + 4x) = −1 − 4 = −5

lim(x→0) (7) = 7 (constant polynomial)

When is something NOT a polynomial?

The moment a variable sits in a denominator (like 1/x) or sits under a root with the variable (√x near 0 from both sides), direct substitution may break. Polynomials are the safe zone: constants, x, x², x³, and anything you can build with +, −, × only.

Standard Form 2 — The Key Formula

Substituting into (xⁿ − aⁿ)/(x − a) always gives 0/0. Instead of factoring from scratch each time, memorise this result — it is the single most examined limit pattern in this chapter.

★ The Key Formula

\lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}

Left side pattern:

numerator = xⁿ − aⁿ (difference of like powers)

denominator = x − a (single linear factor)

Right side recipe:

write the power n, multiply by aⁿ⁻¹

e.g. n = 3, a = 2 → 3 × 2² = 12

n = 2 special case

\lim_{x \to a} \frac{x^2 - a^2}{x - a} = 2a

Factor (x − a)(x + a), cancel, substitute x = a into (x + a) → 2a. E.g. a = 4 → answer 8.

n = 3 special case

\lim_{x \to a} \frac{x^3 - a^3}{x - a} = 3a^2

Factor (x − a)(x² + ax + a²), cancel, substitute → 3a². E.g. a = 2 → 3 × 4 = 12.

Dress code

Denominator must be exactly (x − a) and numerator xⁿ − aⁿ with the same a. Fits? → n·aⁿ⁻¹. Does not fit (e.g. x³ − 27 over x − 2, since 27 ≠ 2³) → factorise manually.

Explore: Standard Form Lab

Drag n and a. The lab shows why naive substitution fails and how n·aⁿ⁻¹ rescues the answer every single time.

Standard Form Lab — choose n and a, watch n·aⁿ⁻¹ appear

Power n = 2

2 (quadratic)5

Point a = 3

Evaluate

lim(x→3) (x² − 3²)/(x − 3)

Naive plug-in

0/0

Formula n · aⁿ⁻¹

2 × 3^1

Answer

How the formula was born

Factor xⁿ − aⁿ = (x − a)(xⁿ⁻¹ + xⁿ⁻²a + … + aⁿ⁻¹). Cancel (x − a), then set x = a: you are adding aⁿ⁻¹ exactly n times → n·aⁿ⁻¹. The formula is just “factor, cancel, count” compressed into one line.

The 0/0 Rescue Flow

Substitution gives 0/0? Run this three-step rescue: factor → cancel → substitute.

Factor → Cancel → Substitute

Step 0 — Direct substitution (the trap)

(x⁴ − 16)/(x − 2) at x = 2 → 2^4 − 2^4 = 0 over 0 → 0/0, blocked

Step 1 — Factorise out (x − 2)

numerator = (x − 2)·(x^3 + x^2·2 + … + 2^3)

Step 2 — Cancel (x − 2) for x ≠ 2

→ (x^3 + x^2·2 + … + 2^3) — the dangerous factor is gone

Step 3 — Substitute x = 2 safely

2^3 + 2^3 + … (4 terms of 2^3) = 4·2^3 = 32

Answer via Key Formula

\lim_{x \to 2} \frac{x^{4} - 2^{4}}{x - 2} = 4 \cdot 2^{3} = 32

Exam Decision Flow — what to do in what order

Is f(x) a polynomial?

Yes → Plug in x = a directly

No → Go to step 2

Does substitution give 0/0?

Yes → Factor / cancel / rationalise

No → Quotient rule (if denominator ≠ 0)

Is it (xⁿ − aⁿ)/(x − a)?

Yes → Answer = n · aⁿ⁻¹ instantly

No → Factor the specific form, cancel, substitute

Solved Examples

Solved Example

Problem

Evaluate lim(x→3) (x² − 9)/(x − 3).

Solution

Solved Example

Problem

Evaluate lim(x→2) (x³ − 8)/(x − 2).

Solution

Solved Example

Problem

Evaluate lim(x→4) (x² + 2x − 8)/(x − 4).

Solution

Solved Example

Problem

Evaluate lim(x→5) (x² − 25)/(x − 5) in one line using the standard form.

Solution

Key Takeaways

Polynomial? Plug in directly: lim(x→a) P(x) = P(a).
★ Key formula: lim(x→a) (xⁿ − aⁿ)/(x − a) = n·aⁿ⁻¹ — one-line answers for difference-of-powers 0/0.
Know the special cases cold: (x² − a²)/(x − a) → 2a and (x³ − a³)/(x − a) → 3a².
Check the dress code: denominator (x − a) and numerator xⁿ − aⁿ with matching a — otherwise factor manually.
Exam order: polynomial → plug in · 0/0 → factor/cancel · key-formula shape → n·aⁿ⁻¹ instantly.