Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
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