Board exam focus — Limits and Derivatives (CBSE & HBSE)

First taste of calculus — intuitive limits, standard limits, and derivative from first principles. CBSE expects 8–10 marks: 1 MCQ + 1 short limit + 1 long derivative-from-first-principles + 1 application of rules.

Intuitive Notion of Limit & Algebra of Limits

Intuitive Idea

The limit of f(x) as x approaches a (written lim_{x→a} f(x)) is the value f(x) gets arbitrarily close to as x gets arbitrarily close to a (but x ≠ a).

The limit exists iff the left-hand limit (LHL, x → a⁻) and right-hand limit (RHL, x → a⁺) are equal.

Symbolic Notation

lim_{x→a⁻} f(x) = LHL, lim_{x→a⁺} f(x) = RHL. If LHL = RHL = L, then lim_{x→a} f(x) = L.

Algebra of Limits

If lim f(x) = ℓ and lim g(x) = m (both finite), then:

• lim (f ± g) = ℓ ± m
• lim (f · g) = ℓ · m
• lim (f/g) = ℓ/m (if m ≠ 0)
• lim (c · f) = c · ℓ for constant c
• lim (fⁿ) = ℓⁿ

Substitution Rule for Polynomial / Rational Functions

For polynomial p(x): lim_{x→a} p(x) = p(a) (direct substitution).

For rational p(x)/q(x) with q(a) ≠ 0: lim = p(a)/q(a) (substitute).

Indeterminate Forms

When substitution gives 0/0 or ∞/∞ (or other indeterminate forms), apply:

1. Factor cancellation (algebraic simplification).
2. Rationalisation (when surds are present).
3. Standard limits (next module).

Common Algebraic Tricks

• Factor and cancel: (x² − a²)/(x − a) → cancel (x − a) → answer = 2a.
• Rationalise: lim_{x→0} (√(x + 1) − 1)/x → multiply by √(x+1) + 1.
• Conjugate for surd-containing limits.

Continuity (Class 12 Preview)

f is continuous at a iff lim_{x→a} f(x) = f(a). All polynomials are continuous everywhere.

CBSE Tip: Always check whether direct substitution is valid before applying tricks — if it works, use it.

Standard Limits

Standard Algebraic Limit

lim_{x→a} (xⁿ − aⁿ) / (x − a) = n · aⁿ⁻¹ (for any rational n).

Standard Trigonometric Limits

• lim_{x→0} (sin x)/x = 1 (x in radians)
• lim_{x→0} (tan x)/x = 1
• lim_{x→0} (1 − cos x)/x = 0
• lim_{x→0} (1 − cos x)/x² = 1/2

Standard Exponential / Logarithmic Limits (preview)

• lim_{x→0} (eˣ − 1)/x = 1
• lim_{x→0} (log(1 + x))/x = 1
• lim_{x→0} (aˣ − 1)/x = log_e a

Strategy for Standard Limits

1. Identify the standard form.
2. Substitute / manipulate the expression to fit the standard form (multiply/divide as needed).
3. Apply the limit value.

Example Using sin x / x

lim_{x→0} sin 3x / x. Write as 3 · (sin 3x / (3x)). As x → 0, 3x → 0 ⇒ (sin 3x)/(3x) → 1. Answer: 3 · 1 = 3.

Example Using xⁿ − aⁿ

lim_{x→2} (x⁵ − 32)/(x − 2). Form = (x⁵ − 2⁵)/(x − 2) ⇒ n · aⁿ⁻¹ = 5 · 2⁴ = 80.

Indeterminate Forms — Checklist

Common forms requiring special handling:

• 0/0, ∞/∞: algebraic / L'Hôpital (Class 12).
• 0 · ∞: rewrite as 0/0 or ∞/∞.
• ∞ − ∞: combine into a single fraction.
• 1^∞, 0⁰, ∞⁰: use logarithm.

Class 11 mostly handles 0/0 algebraically.

HBSE Tip: Always write the standard limit you are using in the working — examiners award marks for identifying the formula.

Derivative — First Principles

Definition (First Principles)

The derivative of f at x is: f′(x) = lim_{h→0} [f(x + h) − f(x)] / h (whenever the limit exists).

Derivative at a Specific Point

f′(a) = lim_{x→a} [f(x) − f(a)] / (x − a).

Geometric Meaning

f′(a) = slope of the tangent to y = f(x) at the point (a, f(a)).

Physical Meaning

If f(t) represents position at time t, then f′(t) = velocity. If f(t) represents velocity, then f′(t) = acceleration.

Notation

f′(x) = dy/dx = d/dx [f(x)] = Df(x).

Computing Derivatives by First Principles

Three-step recipe:

1. Write f(x + h) − f(x).
2. Divide by h and simplify.
3. Take the limit as h → 0.

Sample Derivations

f(x) = x²: f(x + h) − f(x) = (x + h)² − x² = 2xh + h². [f(x + h) − f(x)] / h = 2x + h → 2x as h → 0. So f′(x) = 2x.

f(x) = sin x: f(x + h) − f(x) = sin(x + h) − sin x = 2 cos(x + h/2) sin(h/2). Divide by h: 2 cos(x + h/2) · sin(h/2)/h = cos(x + h/2) · sin(h/2)/(h/2). As h → 0: cos(x) · 1 = cos x. So d/dx (sin x) = cos x.

Differentiability vs Continuity

• Differentiable ⇒ continuous.
• Continuous does NOT imply differentiable (e.g. |x| at x = 0).

Standard Derivative List

CBSE Tip: First-principles questions are mandatory in board exams. Write all three steps (substitution, simplification, limit) for full marks.

Rules of Differentiation & Standard Derivatives

Algebra of Derivatives

For differentiable u(x) and v(x):

1. Sum / Difference: (u ± v)′ = u′ ± v′
2. Constant multiple: (c · u)′ = c · u′
3. Product Rule: (u · v)′ = u′v + uv′
4. Quotient Rule: (u/v)′ = (u′v − uv′)/v² (v ≠ 0)

Chain Rule (Class 12, but used informally in 11)

If y = f(g(x)), dy/dx = f′(g(x)) · g′(x).

Example: y = sin(2x) ⇒ dy/dx = cos(2x) · 2 = 2 cos(2x).

Derivatives of Trigonometric Functions (Summary)

Derivatives of Exponentials & Logs

• d/dx (eˣ) = eˣ
• d/dx (aˣ) = aˣ · log_e a
• d/dx (log_e x) = 1/x
• d/dx (log_a x) = 1/(x · log_e a)

Derivative of Polynomial

For p(x) = aₙxⁿ + … + a₁x + a₀: p′(x) = n aₙ xⁿ⁻¹ + … + a₁.

Higher-Order Derivatives (Preview)

• Second derivative: f″(x) = d/dx [f′(x)].
• Third: f‴(x) and so on.
• Notation: dⁿy/dxⁿ.

Common Derivative Patterns

• d/dx [(ax + b)ⁿ] = n a (ax + b)ⁿ⁻¹.
• d/dx [sin(ax + b)] = a cos(ax + b).
• d/dx [eᵃˣ] = a eᵃˣ.

Applications

• Equation of tangent at (a, f(a)): y − f(a) = f′(a)(x − a).
• Equation of normal: y − f(a) = (−1/f′(a))(x − a).
• Rate of change, velocity, acceleration problems.

HBSE Tip: Always state the rule being applied ("product rule" or "chain rule") in your working — examiners look for it.

Frequently asked questions

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Do these notes follow CBSE and HBSE?

Yes. The Limits and Derivatives notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.

What does the Limits and Derivatives chapter cover?

Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Limits and Derivatives.