Limits and Derivatives Class 11 Notes & Important Questions

Revision notes

Limits and Derivatives: revision notes

1. Limits

$lim_{x \to a} f (x) = L$ 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, $lim_{x \to a} p (x) = p (a)$ .
A rational function with a non-zero denominator at $a$ : substitute.
Form $\frac{0}{0}$ : factorise and cancel the common factor $(x - a)$ , or rationalise surds, then substitute.
$lim_{x \to a} \frac{x^{n} - a^{n}}{x - a} = n a^{n - 1}$ .

3. Trigonometric limits

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

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

$\frac{d}{d x} x^{n} = n x^{n - 1}$ ; $\frac{d}{d x} c = 0$ ; $(c f)^{'} = c f^{'}$ ; $(f \pm g)^{'} = f^{'} \pm g^{'}$ .
Product rule $(u v)^{'} = u^{'} v + u v^{'}$ ; quotient rule ${(\frac{u}{v})}^{'} = \frac{u^{'} v - u v^{'}}{v^{2}}$ .
$\frac{d}{d x} \sin x = \cos x$ , $\frac{d}{d x} \cos x = - \sin x$ , $\frac{d}{d x} \tan x = \sec^{2} x$ .

Worked example 1

Find $lim_{x \to 3} \frac{x^{2} - 9}{x - 3}$ .

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

Worked example 2

Find $lim_{x \to 0} \frac{\sin 4 x}{\sin 2 x}$ .

$\frac{\sin 4 x}{4 x} \cdot \frac{2 x}{\sin 2 x} \cdot 2 \to 2$ .

Worked example 3

Differentiate $y = x^{3} \cos x$ .

$y^{'} = 3 x^{2} \cos x - x^{3} \sin x$ .

Common errors

Writing $\frac{0}{0} = 0$ or $1$ : it means "simplify first".
$\frac{\sin 3 x}{x} \to 1$ : it tends to $3$ .
Product rule applied as $(u v)^{'} = u^{'} v^{'}$ ; quotient rule with the numerator the wrong way round.
$\frac{d}{d x} \cos x = \sin x$ : 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.

Formulas for this chapter: Chapter 12 on the Class 11 formula sheet (free PDF and A3 poster).

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 40 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

$lim_{x \to 2} \frac{x^{2} - 4}{x - 2}$ equals

(a) $0$
(b) $2$
(c)does not exist
(d) $4$

Q2·1 mark·Multiple choiceRules and standard derivatives

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

(a) $1$
(b) $0$
(c) $6$
(d) $- 5$

Q3·1 mark·Multiple choiceTrigonometric limits

$lim_{x \to 0} \frac{\sin 3 x}{x}$ equals

(a) $1$
(b) $\frac{1}{3}$
(c) $3$
(d) $0$

Worked solution·3 marks·Short answerThe derivative

Find the derivative of $f (x) = 3 x^{2} + 1$ from first principles.

Full worked solution Free, no sign-in

$\frac{f (x + h) - f (x)}{h}$ $= \frac{3 (x + h)^{2} + 1 - 3 x^{2} - 1}{h}$ M1
$= \frac{6 x h + 3 h^{2}}{h} = 6 x + 3 h$ A1
$f^{'} (x) = lim_{h \to 0} (6 x + 3 h) = 6 x$ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original calculus and functions problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S128. All 24 →

Knowledge organiser: Limits and Derivatives on one page (key facts, formulas and mistakes to avoid; printable PDF)

Formulas in this topic: Derivative of sin x · Power rule. All formulas A–Z

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, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Preparing for JEE or CUET? Try the JEE and CUET-style questions on limits, continuity and differentiability.

Limits and Derivatives: common questions

What is in CBSE Class 11 Maths Chapter 12, Limits and Derivatives?

The topics are: Polynomial and rational functions, Rules and standard derivatives, Trigonometric limits, The derivative, Limits. This page has revision notes for each, and 40 board-style practice questions with step-marked solutions.

How many marks is Limits and Derivatives worth?

It is in the Calculus unit, which carries 8 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks 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.

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

Limits and Derivatives Class 11: notes and important questions

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