Limits and Derivatives Class 11: Notes, Standard Limits and Solved Questions

What Limits and Derivatives is about

Weightage: This chapter is part of Unit IV, Calculus, which the syllabus gives 8 of the 80 theory marks in the school annual exam (CBSE adds that there is no fixed chapter-wise weightage).Source: CBSE Mathematics (041) syllabus 2026-27, Class XI course structure, Unit IV Calculus

Key concepts

Derivative as rate of change

NCERT section 12.2

Suppose a body falls so that its distance after t seconds is $s(t)=4.9t^2$ metres. Its average velocity from t = 2 to t = 2 + h is
$\frac{s(2+h)-s(2)}{h}=\frac{4.9(4h+h^2)}{h}=19.6+4.9h$

As h shrinks to 0 (from either side), this approaches 19.6 m/s: the velocity at the instant t = 2.

Geometrically, the average velocity is the slope of the chord joining two points of the distance-time graph. As the second point slides towards the first, the chord turns into the tangent, and the slope of the tangent is the instantaneous rate of change.

This 'let h → 0' process is exactly what limits make precise.

Worked example

For $s(t)=4.9t^2$, find the average velocity from t = 2 to t = 2.1, and the instantaneous velocity at t = 2. [2 marks]

Try it first, then show the solutionSolution

1. Average: $19.6+4.9(0.1)=20.09$ m/s
2. Letting h → 0: 19.6 m/s

Answer: 20.09 m/s; 19.6 m/s

Intuitive idea of a limit

NCERT section 12.3

${\lim }_{x\to a}f(x)=L$ means: as x gets closer and closer to a (from both sides, but $x\ne a$), f(x) gets closer and closer to L.

• Left-hand limit (LHL): ${\lim }_{x\to a^{-}}f(x)$, approaching from values less than a.
• Right-hand limit (RHL): ${\lim }_{x\to a^{+}}f(x)$, from values greater than a.
• The limit exists exactly when LHL and RHL both exist and are equal.

The limit does not care about the value f(a), or even whether f(a) is defined. For $f(x)=\frac{x^2-1}{x-1}$, f(1) is undefined but ${\lim }_{x\to 1}f(x)=2$.

Examples where the limit fails: $\frac{|x|}{x}$ at 0 (LHL −1, RHL 1); [x] at an integer n (LHL n − 1, RHL n).

Worked example

Let $f(x)=3x+2$ for $x\le 1$ and $f(x)=x^2+4$ for $x>1$. Find ${\lim }_{x\to 1}f(x)$. Does ${\lim }_{x\to 2}[x]$ exist? [2 marks]

Try it first, then show the solutionSolution

1. LHL $=3(1)+2=5$; RHL $=1^2+4=5$; equal, so the limit is 5
2. Near 2 from the left [x] = 1, from the right [x] = 2: LHL ≠ RHL, so no limit

Answer: 5; does not exist

Algebra of limits; polynomials and rational functions

NCERT section 12.3.1 and 12.3.2

If ${\lim }_{x\to a}f(x)$ and ${\lim }_{x\to a}g(x)$ both exist:

• limit of $f\pm g$ = sum or difference of limits
• limit of $fg$ = product of limits; limit of $kf$ = k × limit
• limit of $\frac{f}{g}$ = quotient of limits, if the limit of g is not 0

Polynomials: ${\lim }_{x\to a}p(x)=p(a)$: just substitute.

Rational functions $\frac{p(x)}{q(x)}$:

• If $q(a)\ne 0$, substitute.
• If $p(a)=q(a)=0$ (the $\frac{0}{0}$ form), $(x-a)$ is a factor of both: factorise, cancel, then substitute. Or rationalise if there are square roots.

Standard limit (Theorem): for any positive integer n,
${\lim }_{x\to a}\frac{x^n-a^n}{x-a}=n{a}^{n-1}$
(it holds for any rational n with a > 0).

Worked example

Evaluate (a) ${\lim }_{x\to 3}\frac{x^2-9}{x^2-5x+6}$ (b) ${\lim }_{x\to 1}\frac{x^{12}-1}{x^8-1}$. [3 marks]

Try it first, then show the solutionSolution

1. (a) $\frac{(x-3)(x+3)}{(x-3)(x-2)}=\frac{x+3}{x-2}\to \frac{6}{1}=6$
2. (b) Divide top and bottom by $x-1$: $\frac{x^{12}-1}{x-1}\to 12$ and $\frac{x^8-1}{x-1}\to 8$
3. Limit $=\frac{12}{8}=\frac{3}{2}$

Answer: 6; $\frac{3}{2}$

Limits of trigonometric functions

NCERT section 12.4

Sandwich theorem: if $f(x)\le g(x)\le h(x)$ near a, and $\lim f=\lim h=L$, then $\lim g=L$.

Using a unit-circle area argument, $\cos x<\frac{\sin x}{x}<1$ for small $x\ne 0$. Both sides tend to 1, so:

• ${\lim }_{x\to 0}\frac{\sin x}{x}=1$ (x in radians)
• ${\lim }_{x\to 0}\frac{1-\cos x}{x}=0$

Consequences:

• ${\lim }_{x\to 0}\frac{\tan x}{x}=1$
• ${\lim }_{x\to 0}\frac{\sin ax}{bx}=\frac{a}{b}$, ${\lim }_{x\to 0}\frac{\sin ax}{\sin bx}=\frac{a}{b}$
• ${\lim }_{x\to 0}\frac{1-\cos ax}{x^2}=\frac{a^2}{2}$ (use $1-\cos ax=2{\sin }^2\frac{ax}{2}$)

Technique: make the angle inside sin match the denominator by multiplying and dividing.

Worked example

Evaluate ${\lim }_{x\to 0}\frac{\sin 4x}{\sin 6x}$ and ${\lim }_{x\to 0}\frac{1-\cos 2x}{x^2}$. [3 marks]

Try it first, then show the solutionSolution

1. $\frac{\sin 4x}{\sin 6x}=\frac{\sin 4x}{4x}\cdot \frac{6x}{\sin 6x}\cdot \frac{4}{6}\to 1\cdot 1\cdot \frac{2}{3}$
2. $1-\cos 2x=2{\sin }^{2}x$, so $\frac{2{\sin }^{2}x}{x^2}=2{\left(\frac{\sin x}{x}\right)}^2$
3. $\to 2\times 1=2$

Answer: $\frac{2}{3}$; 2

Exponential and logarithmic limits

NCERT section Syllabus topic (not in the rationalised NCERT chapter)

The 2026-27 syllabus also lists limits of exponential and logarithmic functions. The two standard results (with e ≈ 2.718 and natural logarithm, ${\log }_e$) are
${\lim }_{x\to 0}\frac{e^x-1}{x}=1,{\lim }_{x\to 0}\frac{{\log }_e(1+x)}{x}=1$

They play the same role as $\frac{\sin x}{x}\to 1$: adjust the expression so the same small quantity appears in the exponent (or inside the log) and in the denominator.

• ${\lim }_{x\to 0}\frac{e^{3x}-1}{x}=3\cdot \lim \frac{e^{3x}-1}{3x}=3$
• ${\lim }_{x\to 0}\frac{{\log }_e(1+2x)}{x}=2$

For continuous functions like $e^x$ and $\log x$ (x > 0), limits at ordinary points are found by substitution: ${\lim }_{x\to 1}{e}^x=e$.

Worked example

Evaluate ${\lim }_{x\to 0}\frac{e^{3x}-1}{x}$ and ${\lim }_{x\to 0}\frac{{\log }_e(1+2x)}{x}$. [2 marks]

Try it first, then show the solutionSolution

1. $\frac{e^{3x}-1}{x}=3\cdot \frac{e^{3x}-1}{3x}\to 3\times 1$
2. $\frac{{\log }_e(1+2x)}{x}=2\cdot \frac{{\log }_e(1+2x)}{2x}\to 2\times 1$

Answer: 3; 2

Derivative from first principle

NCERT section 12.5

The derivative of f at a is
$f^{\prime }(a)={\lim }_{h\to 0}\frac{f(a+h)-f(a)}{h}$
if this limit exists. It is the slope of the tangent to y = f(x) at (a, f(a)).

The derivative function is $f^{\prime }(x)=\frac{d}{dx}f(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$. Finding it straight from this definition is called differentiating from first principle.

Standard results by first principle:

• $\frac{d}{dx}(c)=0$; $\frac{d}{dx}(x)=1$; $\frac{d}{dx}(x^2)=2x$
• $\frac{d}{dx}\left(\frac{1}{x}\right)=-\frac{1}{x^2}$
• $\frac{d}{dx}(\sin x)=\cos x$ (use $\sin (x+h)-\sin x=2\cos (x+\frac{h}{2})\sin \frac{h}{2}$)
• $\frac{d}{dx}(\cos x)=-\sin x$

Worked example

Find the derivative of $f(x)=\frac{1}{x+2}$ from first principle. [3 marks]

Try it first, then show the solutionSolution

1. $\frac{f(x+h)-f(x)}{h}=\frac{1}{h}\left(\frac{1}{x+h+2}-\frac{1}{x+2}\right)=\frac{1}{h}\cdot \frac{(x+2)-(x+h+2)}{(x+2)(x+h+2)}$
2. $=\frac{-1}{(x+2)(x+h+2)}$
3. As h → 0: $f^{\prime }(x)=-\frac{1}{(x+2{)}^2}$

Answer: $-\frac{1}{(x+2{)}^2}$

Algebra of derivatives; polynomials and trigonometric functions

NCERT section 12.5.1 and 12.5.2

For differentiable u and v:

• Sum/difference: $(u\pm v{)}^{\prime }=u^{\prime }\pm v^{\prime }$
• Constant multiple: $(ku{)}^{\prime }=k{u}^{\prime }$
• Product (Leibniz) rule: $(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$
• Quotient rule: ${\left(\frac{u}{v}\right)}^{\prime }=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$, where $v\ne 0$

Power rule: $\frac{d}{dx}{x}^n=n{x}^{n-1}$ for every positive integer n (proved by first principle or by the product rule and induction). So every polynomial is differentiated term by term.

Trigonometric derivatives:

• $(\sin x{)}^{\prime }=\cos x$, $(\cos x{)}^{\prime }=-\sin x$
• $(\tan x{)}^{\prime }={\sec }^{2}x$, $(\cot x{)}^{\prime }=-{\csc }^{2}x$ (quotient rule)
• $(\sec x{)}^{\prime }=\sec x\tan x$, $(\csc x{)}^{\prime }=-\csc x\cot x$

Worked example

Differentiate (a) $(x^2+1)(3x-2)$ (b) $\frac{x+1}{x-1}$ (c) $x\sin x$. [3 marks]

Try it first, then show the solutionSolution

1. (a) $2x(3x-2)+3(x^2+1)=9x^2-4x+3$
2. (b) $\frac{(1)(x-1)-(x+1)(1)}{(x-1{)}^2}=\frac{-2}{(x-1{)}^2}$
3. (c) $1\cdot \sin x+x\cos x=\sin x+x\cos x$

Answer: $9x^2-4x+3$; $-\frac{2}{(x-1{)}^2}$; $\sin x+x\cos x$

Maths in real life: Limits and Derivatives

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Topics in this chapter

• Derivatives: First Principle and Rules of Differentiation
• Evaluating Limits: Algebraic, Trigonometric and Exponential

Limits and Derivatives formula sheet

Standard algebraic limit

${\lim }_{x\to a}\frac{x^n-a^n}{x-a}=n{a}^{n-1}$

Trigonometric limits

${\lim }_{x\to 0}\frac{\sin x}{x}=1,{\lim }_{x\to 0}\frac{1-\cos x}{x}=0,{\lim }_{x\to 0}\frac{\tan x}{x}=1$

Exponential and log limits

${\lim }_{x\to 0}\frac{e^x-1}{x}=1,{\lim }_{x\to 0}\frac{{\log }_e(1+x)}{x}=1$

Derivative (first principle)

$f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$

Product and quotient rules

$(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime },{\left(\frac{u}{v}\right)}^{\prime }=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$

Power and trigonometric derivatives

$(x^n{)}^{\prime }=n{x}^{n-1},(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x,(\tan x{)}^{\prime }={\sec }^{2}x$

Other trigonometric derivatives

$(\cot x{)}^{\prime }=-{\csc }^{2}x,(\sec x{)}^{\prime }=\sec x\tan x,(\csc x{)}^{\prime }=-\csc x\cot x$

How the school exam asks Limits and Derivatives

The question patterns to expect, and the method that works for each.

Tricks to remember Limits and Derivatives

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Zero over zero? Factor and cancel

  If substitution gives $\frac{0}{0}$, (x − a) is hiding in both: find it and cancel.

  Use it when: Rational limits.

• Match the angle

  $\frac{\sin (\text{something})}{\text{same something}}\to 1$: multiply and divide to make them match.

  Use it when: Trigonometric limits.

• Low d-high minus high d-low

  Quotient rule: $\frac{v{u}^{\prime }-u{v}^{\prime }}{v^2}$ ('low d-high minus high d-low, over low squared').

  Use it when: Quotient rule.

• Co-functions get a minus

  Derivatives of cos, cot and cosec all start with a minus sign.

  Use it when: Trigonometric derivatives.

• Bring down and reduce

  $x^n$: bring the power down in front and reduce it by one.

  Use it when: Power rule.

Limits and Derivatives: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section E style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 4-mark question split into 1 + 1 + 1 + 1 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Limits and Derivatives

The slips examiners see most often, and what to write instead.

• ✗Mistake: Writing f(a) as the limit without checking

  Substituting into $\frac{0}{0}$ forms.

  ✓Do this: If substitution gives $\frac{0}{0}$, simplify first.

  Usually costs: Whole question.

• ✗Mistake: Using degrees in $\frac{\sin x}{x}$

  Forgetting the result needs radians.

  ✓Do this: All calculus with trigonometric functions uses radians.

  Usually costs: 1 mark.

• ✗Mistake: Product rule as $(uv{)}^{\prime }=u^{\prime }{v}^{\prime }$

  Guessing a simpler rule.

  Wrong working

  $\frac{d}{dx}(x\sin x)=\cos x$

  Right working

  $\sin x+x\cos x$

  ✓Do this: Two terms: differentiate one factor at a time.

  Usually costs: 1-2 marks.

• ✗Mistake: Wrong order in the quotient rule numerator

  Writing $u{v}^{\prime }-u^{\prime }v$.

  ✓Do this: Denominator's function first: $u^{\prime }v-u{v}^{\prime }$ (derivative of top times bottom first).

  Usually costs: 1 mark (sign error).

• ✗Mistake: Letting h → 0 before cancelling in first-principle work

  Getting $\frac{0}{0}$ and stopping.

  ✓Do this: Simplify the difference quotient so h cancels, then put h = 0.

  Usually costs: 1-2 marks.

• ✗Mistake: Claiming a limit exists when LHL ≠ RHL

  Checking only one side.

  ✓Do this: For piecewise functions always compute both sides.

  Usually costs: 1 mark.

NCERT exercise guide: Limits and Derivatives

What each exercise in the NCERT chapter tests, how to approach it, and a question of the same kind solved step by step. Questions here are written in our own words.

Practice questions on Limits and Derivatives

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   ${\lim }_{x\to 3}(x^2+2x-1)$ is:
   (a) 12
   (b) 14
   (c) 15
   (d) 16

   Hint

   Substitute.

   Answer

   (b) 14

2. Q2 · 1 mark · MCQExam level

   The derivative of $\cot x$ is:
   (a) ${\csc }^{2}x$
   (b) $-{\csc }^{2}x$
   (c) ${\sec }^{2}x$
   (d) $-\csc x\cot x$

   Hint

   Co-functions get a minus.

   Answer

   (b)

3. Q3 · 1 mark · Assertion-reasonExam level

   Assertion (A): $\frac{d}{dx}(x^2\sin x)=2x\cos x$.
   Reason (R): The derivative of a product is the product of the derivatives.
   (a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) Both A and R false.

   Hint

   Product rule.

   Answer

   (d): the correct derivative is $2x\sin x+x^2\cos x$, and R is false.

4. Q4 · 1 mark · Very short answerExam level

   Evaluate ${\lim }_{x\to 2}\frac{x^3-8}{x-2}$.

   Hint

   Standard limit with n = 3.

   Answer

   $3\times 2^2=12$.

5. Q5 · 2 marks · Very short answerHarder exam level

   Evaluate ${\lim }_{x\to 0}\frac{\sin 3x+4x}{2x+\sin x}$.

   Hint

   Divide top and bottom by x.

   Answer

   Dividing top and bottom by x: $\frac{3+4}{2+1}=\frac{7}{3}$.

6. Q6 · 2 marks · Very short answerHarder exam level

   Find the derivative of $x^2+3x$ at x = 2 from first principle.

   Hint

   $\frac{f(2+h)-f(2)}{h}$.

   Answer

   $\frac{(2+h{)}^2+3(2+h)-10}{h}=\frac{7h+h^2}{h}=7+h\to 7$.

7. Q7 · 2 marks · Very short answerHarder exam level

   Differentiate $(x+a)(x^2-b)$ for constants a and b.

   Hint

   Product rule or expand.

   Answer

   $(x^2-b)+(x+a)(2x)=3x^2+2ax-b$.

8. Q8 · 3 marks · Short answerHarder exam level

   Differentiate $\frac{x}{1+\tan x}$.

   Hint

   Quotient rule; $(\tan x{)}^{\prime }={\sec }^{2}x$.

   Answer

   $\frac{1\cdot (1+\tan x)-x{\sec }^{2}x}{(1+\tan x{)}^2}=\frac{1+\tan x-x{\sec }^{2}x}{(1+\tan x{)}^2}$.

9. Q9 · 3 marks · Short answerHarder exam level

   Find the derivative of $\cos (x+\frac{\pi }{6})$ from first principle.

   Hint

   Use cos A − cos B.

   Answer

   $\frac{\cos (x+h+\frac{\pi }{6})-\cos (x+\frac{\pi }{6})}{h}=\frac{-2\sin (x+\frac{\pi }{6}+\frac{h}{2})\sin \frac{h}{2}}{h}\to -\sin (x+\frac{\pi }{6})$.

10. Q10 · 2 marks · Very short answerHOTS / competency

    Evaluate ${\lim }_{x\to 0}\frac{e^{2x}-1}{\sin 3x}$.

    Hint

    Split into two standard forms.

    Answer

    $\frac{e^{2x}-1}{2x}\cdot \frac{3x}{\sin 3x}\cdot \frac{2}{3}\to \frac{2}{3}$.

11. Q11 · 3 marks · Short answerHOTS / competency

    Show that the function $f(x)=|x|$ has no derivative at x = 0 by computing the two one-sided limits of $\frac{f(0+h)-f(0)}{h}$.

    Hint

    Sign of h.

    Answer

    $\frac{|h|}{h}$ is 1 for h > 0 and −1 for h < 0. The one-sided limits (1 and −1) differ, so the limit defining f′(0) does not exist: the graph has a corner at the origin.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.When does ${\lim }_{x\to a}f(x)$ exist?Show answer

   When the left-hand and right-hand limits exist and are equal.

2. 2.${\lim }_{x\to 2}\frac{x^5-32}{x-2}$?Show answer

   $5\times 2^4=80$.

3. 3.${\lim }_{x\to 0}\frac{\sin 5x}{3x}$?Show answer

   $\frac{5}{3}$.

4. 4.Define f′(x) by first principle.Show answer

   ${\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$.

5. 5.Derivative of $\cos x$?Show answer

   $-\sin x$.

6. 6.Derivative of $\tan x$?Show answer

   ${\sec }^{2}x$.

7. 7.State the product rule.Show answer

   $(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$.

8. 8.What does f′(a) mean on a graph?Show answer

   The slope of the tangent at (a, f(a)).

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you explain a limit in your own words?
• ☐Can you find LHL and RHL of a piecewise function?
• ☐Can you evaluate $\frac{0}{0}$ limits by factorising or rationalising?
• ☐Can you use $\lim \frac{x^n-a^n}{x-a}=n{a}^{n-1}$?
• ☐Can you prove ${\lim }_{x\to 0}\frac{\sin x}{x}=1$ using the sandwich theorem?
• ☐Can you evaluate limits like $\frac{\sin ax}{\sin bx}$ and $\frac{1-\cos ax}{x^2}$?
• ☐Can you use the exponential and log standard limits?
• ☐Can you differentiate $x^2$, $\frac{1}{x}$ and $\sin x$ from first principle?
• ☐Can you apply the product and quotient rules correctly?
• ☐Can you find velocity or marginal cost as a derivative?

Limits and Derivatives: frequently asked questions