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Limits and Derivatives – Class 11 Mathematics notes, important questions and MCQs

NCERT Class 11 Mathematics, chapter 12. Notes, key terms, important questions with answers and MCQs.

Chapter overview

Limits describe the value a function approaches. Derivatives, defined through limits, measure instantaneous rate of change.

Limits

lim(x→a) f(x) exists if left and right limits are equal.
lim(x→a) (xⁿ − aⁿ)/(x − a) = naⁿ⁻¹.
lim(x→0) sin x/x = 1; lim(x→0) (1 − cos x)/x = 0.

Derivatives

f′(x) = lim(h→0) [f(x + h) − f(x)]/h (first principle).
d/dx xⁿ = nxⁿ⁻¹; d/dx sin x = cos x; d/dx cos x = −sin x.
Product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v².

Key terms

Limit: value a function approaches as x approaches a point
Derivative: instantaneous rate of change of a function
First principle: finding a derivative directly from f′(x) = lim(h→0) [f(x + h) − f(x)]/h

Formulas

lim(x→a) (xⁿ − aⁿ)/(x − a) = naⁿ⁻¹
lim(x→0) sin x/x = 1
d/dx xⁿ = nxⁿ⁻¹

Important questions with answers

Evaluate lim(x→2) (x³ − 8)/(x − 2).2 marks

Using lim (xⁿ − aⁿ)/(x − a) = naⁿ⁻¹: 3 × 2² = 12.

Find the derivative of f(x) = x² from first principles.3 marks

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

Find the derivative of sin x from first principles, and evaluate lim(x→0) (1 − cos 2x)/x².5 marks

f′(x) = lim(h→0) [sin(x + h) − sin x]/h = lim(h→0) 2 cos(x + h/2) sin(h/2)/h = lim cos(x + h/2) × [sin(h/2)/(h/2)] = cos x × 1 = cos x. Limit: 1 − cos 2x = 2 sin²x, so (1 − cos 2x)/x² = 2(sin x/x)² → 2.

Quick revision

sin x/x → 1 as x → 0.
Derivative of a constant is 0.
Left limit = right limit for existence.

Exam tips

For 0/0 forms, factorise and cancel before substituting.
For limits of the form 0/0 with trig functions, rewrite to use lim(x→0) sin x/x = 1 or lim(x→0) tan x/x = 1.

Practice MCQs

lim(x→2) (x + 3) equals
1. 2
2. 3
3. 5
4. 6
Show answer
Answer: (c) 5. Substitute.
d/dx (x³) is
1. x²
2. x³/3
3. 3x
4. 3x²
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Answer: (d) 3x². Power rule.
lim(x→0) sin 3x/x equals
1. 1
2. 1/3
3. 3
4. 0
Show answer
Answer: (c) 3. 3 × (sin 3x/3x).
lim(x→3) (x² − 9)/(x − 3) equals
1. 3
2. 0
3. 9
4. 6
Show answer
Answer: (d) 6. x + 3.
lim(x→0) tan 2x/x equals
1. 1
2. 0
3. 2
4. 1/2
Show answer
Answer: (c) 2. 2 × tan 2x/2x.
d/dx (x sin x) is
1. x cos x
2. sin x + x cos x
3. cos x
4. sin x − x cos x
Show answer
Answer: (b) sin x + x cos x. Product rule.
lim(x→1) (x⁵ − 1)/(x − 1) equals
1. 1
2. 4
3. 0
4. 5
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Answer: (d) 5. n aⁿ⁻¹ = 5.
Derivative of (x² + 1)/x is
1. 1 − 1/x²
2. 1 + 1/x²
3. 2x − 1/x²
4. 2x
Show answer
Answer: (a) 1 − 1/x². d/dx (x + 1/x).

All chapters of Class 11 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.