Chapter Introduction
Limits and Derivatives is Chapter 12 of CBSE Class 11 Mathematics, falling under the Calculus unit. This is your first encounter with calculus — the mathematics of change. The concept of a limit lets you understand what happens to a function as its input approaches a value, and the derivative measures the instantaneous rate of change. This chapter typically carries 8-10 marks in board exams and is the single most important foundation for Class 12 calculus and all competitive exams.
The two key ideas: (1) A limit describes the value a function approaches, even if it never actually reaches that value. (2) A derivative is the slope of the tangent to a curve at a point — it is defined as a limit.
Key Definitions
| Term | Definition |
|---|---|
| Limit | The value L that f(x) approaches as x approaches a, written lim(x->a) f(x) = L |
| Left-Hand Limit (LHL) | The value f(x) approaches as x approaches a from values less than a: lim(x->a-) f(x) |
| Right-Hand Limit (RHL) | The value f(x) approaches as x approaches a from values greater than a: lim(x->a+) f(x) |
| Existence of Limit | A limit exists if and only if LHL = RHL (both exist and are equal) |
| Derivative | The instantaneous rate of change of a function at a point; the slope of the tangent |
| First Principle | Computing the derivative directly from its limit definition: f'(x) = lim(h->0) [f(x+h) - f(x)]/h |
| Differentiability | A function is differentiable at a point if its derivative exists at that point |
Detailed Concepts
1. Intuitive Idea of Limits
Consider f(x) = (x^2 - 1)/(x - 1). At x = 1, this is 0/0 (undefined). But as x gets closer to 1:
- x = 0.9: f(x) = (0.81 - 1)/(0.9 - 1) = -0.19/(-0.1) = 1.9
- x = 0.99: f(x) = 1.99
- x = 1.01: f(x) = 2.01
- x = 1.1: f(x) = 2.1
As x approaches 1, f(x) approaches 2. We write: lim(x->1) (x^2 - 1)/(x - 1) = 2.
This works because (x^2 - 1)/(x - 1) = (x + 1)(x - 1)/(x - 1) = x + 1 for x != 1, and lim(x->1) (x + 1) = 2.
2. Left-Hand and Right-Hand Limits
LHL: lim(x->a-) f(x) = lim(h->0) f(a - h)
RHL: lim(x->a+) f(x) = lim(h->0) f(a + h)
Limit exists if and only if LHL = RHL.
| For piecewise functions or functions involving | x | , you must always check LHL and RHL separately. |
|---|
3. Algebra of Limits
If lim f(x) = L and lim g(x) = M, then:
| Operation | Rule |
|---|---|
| Sum | lim [f(x) + g(x)] = L + M |
| Difference | lim [f(x) - g(x)] = L - M |
| Product | lim [f(x) g(x)] = L M |
| Quotient | lim [f(x)/g(x)] = L/M, provided M != 0 |
| Constant multiple | lim [k f(x)] = k L |
| Power | lim [f(x)]^n = L^n |
4. Limits of Polynomials and Rational Functions
Polynomial: lim(x->a) p(x) = p(a). Simply substitute the value.
Rational function: lim(x->a) p(x)/q(x) = p(a)/q(a), provided q(a) != 0.
If q(a) = 0 but p(a) != 0: the limit does not exist (function goes to infinity).
If p(a) = 0 and q(a) = 0: factor and cancel the common factor (x - a), then substitute.
5. Standard Trigonometric Limits
| Limit | Value |
|---|---|
| lim(x->0) sin(x)/x | 1 |
| lim(x->0) (1 - cos(x))/x | 0 |
| lim(x->0) tan(x)/x | 1 |
| lim(x->0) sin(ax)/bx | a/b |
| lim(x->0) tan(ax)/bx | a/b |
| lim(x->0) (1 - cos(ax))/x^2 | a^2/2 |
Technique for trigonometric limits: Always try to convert the expression into a form involving sin(x)/x or (1 - cos(x))/x, then apply the standard limits.
6. Derivatives — First Principle
The derivative of f(x) at x is defined as:
f'(x) = lim(h->0) [f(x + h) - f(x)]/h
This is called differentiation from first principles (or ab initio, or by definition).
Steps: 1. Replace x with (x + h) in f(x) to get f(x + h). 2. Compute f(x + h) - f(x). 3. Divide by h. 4. Take the limit as h approaches 0. 5. Simplify to get f'(x).
7. Algebra of Derivatives
If f and g are differentiable functions:
| Operation | Rule |
|---|---|
| Sum rule | (f + g)' = f' + g' |
| Difference rule | (f - g)' = f' - g' |
| Product rule | (fg)' = f'g + fg' |
| Quotient rule | (f/g)' = (f'g - fg')/g^2, g != 0 |
| Constant multiple | (kf)' = k * f' |
8. Derivatives of Standard Functions
| Function f(x) | Derivative f'(x) |
|---|---|
| constant c | 0 |
| x^n | nx^{n-1} |
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| tan(x) | sec^2(x) |
| sec(x) | sec(x)tan(x) |
| cosec(x) | -cosec(x)cot(x) |
| cot(x) | -cosec^2(x) |
Standard Limits — Quick Reference Table
| Limit Expression | Value | Condition |
|---|---|---|
| lim(x->0) sin(x)/x | 1 | x in radians |
| lim(x->0) (1 - cos(x))/x | 0 | x in radians |
| lim(x->0) tan(x)/x | 1 | x in radians |
| lim(x->0) sin(ax)/(ax) | 1 | a != 0, x in radians |
| lim(x->0) (1 - cos(ax))/x^2 | a^2/2 | x in radians |
| lim(x->a) p(x) | p(a) | p is polynomial |
| lim(x->a) p(x)/q(x) | p(a)/q(a) | q(a) != 0 |
| lim(x->0) (e^x - 1)/x | 1 | not in NCERT Class 11 but useful |
| lim(x->0) ln(1+x)/x | 1 | not in NCERT Class 11 but useful |
Derivative Rules — Quick Reference Table
| Rule | Formula |
|---|---|
| Power rule | d/dx (x^n) = n * x^{n-1} |
| Constant rule | d/dx (c) = 0 |
| Sum rule | d/dx (f + g) = f' + g' |
| Product rule | d/dx (fg) = f'g + fg' |
| Quotient rule | d/dx (f/g) = (f'g - fg')/g^2 |
| Chain rule (preview) | d/dx f(g(x)) = f'(g(x)) * g'(x) |
Important Numericals
Example 1: Evaluating a limit (0/0 form)
Find lim(x->2) (x^3 - 8)/(x - 2).
Factor: x^3 - 8 = (x - 2)(x^2 + 2x + 4)
lim(x->2) (x - 2)(x^2 + 2x + 4)/(x - 2) = lim(x->2) (x^2 + 2x + 4) = 4 + 4 + 4 = 12
Example 2: Trigonometric limit
Find lim(x->0) sin(3x)/5x.
lim(x->0) sin(3x)/5x = (3/5) lim(x->0) sin(3x)/(3x) = (3/5) 1 = 3/5
Example 3: Checking existence of limit
| Does lim(x->0) | x | /x exist? |
|---|
| LHL: lim(x->0-) | x | /x = lim(x->0-) (-x)/x = -1 |
|---|
| RHL: lim(x->0+) | x | /x = lim(x->0+) x/x = 1 |
|---|
Since LHL = -1 != 1 = RHL, the limit does not exist.
Example 4: Derivative by first principle
Find the derivative of f(x) = x^2 - 3x + 2 by first principle.
f'(x) = lim(h->0) [f(x+h) - f(x)]/h
f(x+h) = (x+h)^2 - 3(x+h) + 2 = x^2 + 2xh + h^2 - 3x - 3h + 2
f(x+h) - f(x) = 2xh + h^2 - 3h = h(2x + h - 3)
[f(x+h) - f(x)]/h = 2x + h - 3
f'(x) = lim(h->0) (2x + h - 3) = 2x - 3
Example 5: Using derivative rules
Find the derivative of f(x) = x^3 * sin(x).
Using product rule: f'(x) = 3x^2 sin(x) + x^3 cos(x)
Example 6: Quotient rule
Find the derivative of f(x) = sin(x)/x^2.
Using quotient rule: f'(x) = (cos(x) x^2 - sin(x) 2x)/x^4 = (x cos(x) - 2 sin(x))/x^3
Common Mistakes to Avoid
1. Assuming limit exists without checking LHL = RHL: For piecewise functions, absolute value functions, and functions with domain restrictions, always compute LHL and RHL separately before concluding.
2. Not simplifying 0/0 forms: If direct substitution gives 0/0, do not conclude the limit does not exist. Factor, rationalize, or use standard limits to simplify first.
3. Degrees in trigonometric limits: The standard limits lim(x->0) sin(x)/x = 1 require x in radians, not degrees. If the problem uses degrees, convert first: x degrees = (pi/180) * x radians.
4. First principle errors: When computing f(x+h), expand carefully. Common errors: forgetting to substitute x+h in every occurrence, or making algebra mistakes in the expansion.
5. Sign errors in derivative of cos(x): d/dx cos(x) = -sin(x). The negative sign is frequently forgotten.
6. Product rule vs chain rule: In Class 11, you use the product rule (fg)' = f'g + fg'. The chain rule is for composite functions and is covered in Class 12, but students sometimes confuse the two.
7. Forgetting to take the limit: In first principle problems, do not stop after dividing by h. You must take the limit h->0 of the simplified expression.
Exam Tips
- For 1-mark questions, know: standard limits (sin(x)/x = 1), derivative of sin, cos, tan, and the power rule.
- For 3-mark questions, practice: evaluating limits (0/0 forms, trigonometric), derivatives by rules, and checking LHL/RHL.
- For 5-mark questions, practice: derivative by first principle (most commonly asked for x^2, 1/x, sin(x)), product and quotient rule applications.
- In first principle questions, write all four steps clearly: (1) write f(x+h), (2) subtract f(x), (3) divide by h, (4) take limit. This ensures partial marks.
- For trigonometric limits, identify the standard limit form and adjust coefficients to match it.
- When a limit does not exist, explicitly state "LHL != RHL" with both values.
Quick Revision Notes
NCERT Exercise Problem Patterns
| Exercise | Topic | Typical Question Type |
|---|---|---|
| 12.1 | Limits, LHL, RHL | Evaluate limits, check existence |
| 12.2 | Trigonometric limits, algebra of limits | sin(x)/x type, rational limits |
| 12.3 | Derivative by first principle | Find f'(x) from definition |
| 12.4 | Derivative rules | Product, quotient, polynomial derivatives |
Derivations to Remember
Derivation 1: lim(x->0) sin(x)/x = 1 (geometric proof)
Consider a unit circle with angle x at centre O. Let P be on the circle, and the tangent at (1,0) meet OP extended at T.
Area of triangle OAP < Area of sector OAP < Area of triangle OAT
(1/2)sin(x)cos(x) < (1/2)x < (1/2)tan(x)
Dividing by (1/2)sin(x): cos(x) < x/sin(x) < 1/cos(x)
Taking reciprocals: 1/cos(x) > sin(x)/x > cos(x)
As x -> 0, cos(x) -> 1, so by squeeze theorem: lim(x->0) sin(x)/x = 1
Derivation 2: Derivative of sin(x) by first principle
f'(x) = lim(h->0) [sin(x+h) - sin(x)]/h
= lim(h->0) [2cos((2x+h)/2)sin(h/2)]/h (using sum-to-product formula)
= lim(h->0) cos(x + h/2) * lim(h->0) sin(h/2)/(h/2)
= cos(x) * 1 = cos(x)
Derivation 3: Derivative of x^n by first principle
f'(x) = lim(h->0) [(x+h)^n - x^n]/h
Using binomial expansion: (x+h)^n = x^n + nx^{n-1}h + n(n-1)/2 * x^{n-2}h^2 + ...
(x+h)^n - x^n = nx^{n-1}h + terms with h^2 and higher
Dividing by h: nx^{n-1} + terms with h
As h -> 0: f'(x) = nx^{n-1}
Additional Solved Examples
Example 7: Rational limit by rationalization
Find lim(x->4) (sqrt(x) - 2)/(x - 4).
Rationalize the numerator: multiply by (sqrt(x) + 2)/(sqrt(x) + 2)
= lim(x->4) (x - 4)/[(x - 4)(sqrt(x) + 2)] = lim(x->4) 1/(sqrt(x) + 2) = 1/4
Example 8: Trigonometric limit with transformation
Find lim(x->0) (sin(5x))/(sin(3x)).
= lim(x->0) [sin(5x)/(5x)] [3x/sin(3x)] (5x/3x)
= 1 1 5/3 = 5/3
Example 9: Derivative of tan(x) by quotient rule
f(x) = sin(x)/cos(x)
f'(x) = [cos(x)cos(x) - sin(x)(-sin(x))]/cos^2(x)
= [cos^2(x) + sin^2(x)]/cos^2(x) = 1/cos^2(x) = sec^2(x)
Practice Checklist
- Evaluate 5 polynomial/rational limits (including 0/0 forms)
- Solve 4 trigonometric limit problems
- Check LHL = RHL for 2 piecewise/absolute value functions
- Find derivatives by first principle for 3 functions
- Apply product rule in 3 problems and quotient rule in 3 problems
- Verify your derivative answers using alternative methods
Quick Revision Notes
1. Limit exists: LHL = RHL (both must exist and be equal) 2. LHL: lim(h->0) f(a - h); RHL: lim(h->0) f(a + h) 3. Polynomial limit: substitute directly; rational: factor and cancel if 0/0 4. Standard limits: sin(x)/x = 1, (1-cos(x))/x = 0, tan(x)/x = 1 (all as x->0, in radians) 5. Derivative (first principle): f'(x) = lim(h->0) [f(x+h) - f(x)]/h 6. Power rule: d/dx(x^n) = n*x^{n-1} 7. Sum rule: (f+g)' = f' + g'; Product rule: (fg)' = f'g + fg' 8. Quotient rule: (f/g)' = (f'g - fg')/g^2 9. d/dx sin(x) = cos(x); d/dx cos(x) = -sin(x); d/dx tan(x) = sec^2(x) 10. For 0/0 limits: factor, rationalize, or use standard limits — never leave as 0/0 11. Always use radians for trigonometric limits 12. Write all steps in first principle — do not skip algebraic simplification