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Class 11 · Chapter 12 · Calculus unit (8 of 80 marks)

Limits and Derivatives Class 11: notes and important questions

Revision notes, 17 board-style questions with the step marks shown, and a four-step route from the basics to 95+, each step ending in a short checkpoint.

  • 17 questions
  • 5 multiple choice, 1 assertion–reason, 3 very short answer, 4 short answer, 2 long answer, 2 case study
  • About 16 hours to master

Calculus unit: 8 of 80 theory marks (Limits and Derivatives).

Revision notes

Limits and Derivatives: revision notes

1. Limits

\(\displaystyle\lim_{x \to a} f(x) = L\) means \(f(x)\) gets as close as we like to \(L\) as \(x\) gets close to \(a\) (\(x \neq a\)). The limit exists when the left-hand limit equals the right-hand limit. Limits of sums, differences, products and quotients (non-zero denominator) are the sums, differences, products and quotients of the limits.

2. Polynomial and rational functions

  • A polynomial: substitute, \(\displaystyle\lim_{x \to a} p(x) = p(a)\).
  • A rational function with a non-zero denominator at \(a\): substitute.
  • Form \(\dfrac00\): factorise and cancel the common factor \((x - a)\), or rationalise surds, then substitute.
  • \(\displaystyle\lim_{x \to a} \frac{x^n - a^n}{x - a} = na^{n-1}\).

3. Trigonometric limits

\(\displaystyle\lim_{x \to 0} \frac{\sin x}{x} = 1\) and \(\displaystyle\lim_{x \to 0} \frac{1 - \cos x}{x} = 0\) (\(x\) in radians). Hence \(\displaystyle\lim_{x \to 0} \frac{\sin ax}{x} = a\) and \(\displaystyle\lim_{x \to 0} \frac{\tan x}{x} = 1\).

4. The derivative

\[f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\]

It is the rate of change of \(f\), and the slope of the tangent to \(y = f(x)\). Finding it from this definition is "from first principles".

5. Rules and standard derivatives

  • \(\dfrac{d}{dx}x^n = nx^{n-1}\); \(\dfrac{d}{dx}c = 0\); \((cf)' = cf'\); \((f \pm g)' = f' \pm g'\).
  • Product rule \((uv)' = u'v + uv'\); quotient rule \(\left(\dfrac uv\right)' = \dfrac{u'v - uv'}{v^2}\).
  • \(\dfrac{d}{dx}\sin x = \cos x\), \(\dfrac{d}{dx}\cos x = -\sin x\), \(\dfrac{d}{dx}\tan x = \sec^2 x\).

Worked example 1

Find \(\displaystyle\lim_{x \to 3} \frac{x^2 - 9}{x - 3}\).

\(\dfrac{(x - 3)(x + 3)}{x - 3} = x + 3 \to 6\).

Worked example 2

Find \(\displaystyle\lim_{x \to 0} \frac{\sin 4x}{\sin 2x}\).

\(\dfrac{\sin 4x}{4x} \cdot \dfrac{2x}{\sin 2x} \cdot 2 \to 2\).

Worked example 3

Differentiate \(y = x^3\cos x\).

\(y' = 3x^2\cos x - x^3\sin x\).

Common errors

  • Writing \(\dfrac00 = 0\) or \(1\): it means "simplify first".
  • \(\dfrac{\sin 3x}{x} \to 1\): it tends to \(3\).
  • Product rule applied as \((uv)' = u'v'\); quotient rule with the numerator the wrong way round.
  • \(\dfrac{d}{dx}\cos x = \sin x\): the sign is negative.

Exam tips

  • Write "\(\lim\)" on every line until the limit is taken.
  • In first-principles proofs, show the expansion of \(f(x + h)\), the cancellation and the limit as separate steps.

Topics in this chapter: Polynomial and rational functions · Rules and standard derivatives · Trigonometric limits · The derivative · Limits.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can evaluate standard limits by substitution or factorising and differentiate polynomials and sin, cos.

Read first: 1. Limits; 2. Polynomial and rational functions; 5. Rules 5 practice questions · checkpoint: 3 questions, 3 marks, pass 80%
Practise step 1
Step 2

Exam standard

You can use sin x/x → 1, differentiate from first principles and apply the product and quotient rules.

Read first: 3. Trigonometric limits; 4. The derivative; 5. Product and quotient rules; Worked examples 1-3 6 practice questions · checkpoint: 3 questions, 8 marks, pass 80%
Practise step 2
Step 3

Full marks on long answers

You can handle limits needing surds or trigonometric rewriting and use derivatives as rates of change.

Read first: 2. Rationalising; 3. Trigonometric limits; 4. Rate of change 3 practice questions · checkpoint: 2 questions, 9 marks, pass 80%
Practise step 3
Step 4

95+ stretch (HOTS)

You can find constants that make a limit exist and prove standard derivatives from the definition.

Read first: 1. Left- and right-hand limits; 4. First principles; Common errors 3 practice questions · checkpoint: 3 questions, 9 marks, pass 80%
Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 2 of the 17 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choicePolynomial and rational functions

\(\displaystyle\lim_{x \to 2} \frac{x^2 - 4}{x - 2}\) equals

  1. (a)\(0\)
  2. (b)\(2\)
  3. (c)does not exist
  4. (d)\(4\)
Q2·1 mark·Multiple choiceRules and standard derivatives

If \(f(x) = 3x^2 - 5x + 2\), then \(f'(1)\) is

  1. (a)\(1\)
  2. (b)\(0\)
  3. (c)\(6\)
  4. (d)\(-5\)
Q3·1 mark·Multiple choiceTrigonometric limits

\(\displaystyle\lim_{x \to 0} \frac{\sin 3x}{x}\) equals

  1. (a)\(1\)
  2. (b)\(\dfrac13\)
  3. (c)\(3\)
  4. (d)\(0\)

Where marks are lost in Limits and Derivatives

  • 0/0 treated as a value. Fix: factorise, cancel or rationalise, then substitute.
  • sin(ax)/x taken to 1. Fix: rewrite as a · sin(ax)/(ax) → a.
  • Product or quotient rule misremembered. Fix: (uv)' = u'v + uv'; (u/v)' = (u'v − uv')/v².
  • Sign of the derivative of cos x lost. Fix: (cos x)' = −sin x.
  • 'lim' dropped from the working, or the limit taken before h is cancelled. Fix: cancel h first, then let h → 0.

Test Limits and Derivatives against the clock

Want it against the clock? Take a timed 30-mark chapter test on Limits and Derivatives, new questions each time (CBSE Essentials or the free trial).