Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
1. How to Use This Page
This page contains complete exercise solutions for Chapter 9 — Differential Calculus: Limits and Continuity of the Tamil Nadu State Board Class 11 Mathematics textbook. Solve each exercise on your own first, then check your working against the solutions below. Worked numerical examples show every step so you can trace the logic from start to finish.
2. Key Concepts (Quick Recap)
Limit Laws
| Rule | Formula |
|---|---|
| Sum/Difference | lim[f(x) ± g(x)] = lim f(x) ± lim g(x) |
| Product | lim[f(x) · g(x)] = lim f(x) · lim g(x) |
| Quotient | lim[f(x)/g(x)] = lim f(x) / lim g(x), lim g(x) ≠ 0 |
| Constant multiple | lim[c · f(x)] = c · lim f(x) |
| Composition | lim[f(g(x))] = f(lim g(x)) if f is continuous |
Standard Limits
Continuity at x = a
A function f is continuous at x = a if and only if: 1. f(a) is defined 2. lim(x→a) f(x) exists (LHL = RHL) 3. lim(x→a) f(x) = f(a)
Types of Discontinuity
| Type | Description |
|---|---|
| Removable | Limit exists but ≠ f(a) or f(a) undefined |
| Jump | LHL ≠ RHL (finite jump) |
| Infinite | Limit tends to ±∞ |
| Oscillatory | Limit does not exist (oscillates) |
3. I. Choose the Best Answer (MCQs)
Q1. lim(x→0) (sin 5x)/x equals:
(a) 0   (b) 1   (c) 5   (d) 1/5
Q2. lim(x→0) (1 − cos 2x)/x² equals:
(a) 0   (b) 1   (c) 2   (d) 4
| Q3. The function f(x) = | x | at x = 0 is: |
|---|
(a) Discontinuous   (b) Continuous but not differentiable   (c) Both continuous and differentiable   (d) None of these
| Explanation: f(x) = | x | is continuous everywhere. At x = 0, LHL = RHL = 0 = f(0). But the left derivative is −1 and right derivative is 1, so not differentiable. |
|---|
Q4. lim(x→∞) (1 + 3/x)ˣ equals:
(a) 1   (b) e   (c) e³   (d) ∞
Q5. If f(x) = {x² − 1 for x ≠ 1; 2 for x = 1}, then f is ___ at x = 1.
(a) Continuous   (b) Removable discontinuity   (c) Jump discontinuity   (d) Infinite discontinuity
4. II. Fill in the Blanks
1. lim(x→0) (sin x)/x = 1.
2. lim(x→0) (tan x)/x = 1.
3. If lim(x→a⁻) f(x) = lim(x→a⁺) f(x), then the limit at x = a exists.
4. lim(x→0) (eˣ − 1)/x = 1.
5. lim(x→0) ((1 + x)ⁿ − 1)/x = n.
6. A function f is continuous at x = a if lim(x→a) f(x) = f(a).
7. lim(x→0) (1 − cos x)/x² = 1/2.
8. The type of discontinuity where LHL ≠ RHL is called jump discontinuity.
9. lim(x→∞) (1 + 1/x)ˣ = e.
| 10. lim(x→0) x sin(1/x) = 0 (by squeeze theorem since − | x | ≤ x sin(1/x) ≤ | x | ). |
|---|
11. If f(x) = [x] (greatest integer function), then f is discontinuous at every integer value of x.
12. lim(x→0) (ln(1 + x))/x = 1.
5. III. True or False (with Corrections)
1. Statement: If lim(x→a) f(x) exists, then f is continuous at x = a. Answer: False. The limit must also equal f(a) for continuity. The limit could exist but f(a) could be undefined or different.
2. Statement: Every polynomial function is continuous everywhere. Answer: True. Polynomials are continuous for all real x.
3. Statement: lim(x→0) sin(1/x) = 0. Answer: False. sin(1/x) oscillates between −1 and 1 as x → 0, so the limit does not exist. However, lim(x→0) x·sin(1/x) = 0 by the squeeze theorem.
4. Statement: A rational function is continuous everywhere. Answer: False. A rational function p(x)/q(x) is discontinuous where q(x) = 0.
5. Statement: The greatest integer function [x] is continuous at every non-integer point. Answer: True. [x] is continuous on (n, n+1) for each integer n, but discontinuous at every integer.
6. IV. Match the Following
| Column A | Column B |
|---|---|
| 1. lim(x→0)(sin x)/x | a. eⁿ |
| 2. lim(x→0)(eˣ−1)/x | b. 1 |
| 3. lim(x→∞)(1 + n/x)ˣ | c. 1 |
| 4. lim(x→0)(1 − cos x)/x² | d. 0 |
| 5. lim(x→0) x² | e. 1/2 |
| 6. lim(x→0)(tan x)/x | f. 1 |
7. V. Answer Briefly (2-Mark Questions)
Q1. Evaluate lim(x→2) (x³ − 8)/(x − 2).
Q2. Evaluate lim(x→0) (sin 3x)/(sin 5x).
Q3. Check continuity of f(x) = 2x + 3 at x = 1.
Q4. Find lim(x→0) (tan x − sin x)/x³.
8. VI. Answer in Detail (5-Mark Questions)
Q1. Evaluate lim(x→0) (1 − cos 2x)/(x sin 3x).
Q2. Check the continuity of f(x) at x = 2 where: f(x) = {x² + 3x + 2 for x ≠ 2; 10 for x = 2}
Q3. Evaluate lim(x→0) (tan 5x − tan 3x)/(sin 3x).
| Q4. Discuss the continuity of f(x) = | x | /x at x = 0. |
|---|
9. Common Mistakes to Avoid
1. Stopping at 0/0: Getting 0/0 does NOT mean the limit is undefined. It means you need to simplify — factorise, rationalise, or use standard limits.
2. Forgetting to check both LHL and RHL: For continuity, you must verify lim(x→a⁻) f(x) = lim(x→a⁺) f(x) = f(a). Only checking one side is insufficient.
3. Confusing limit existence with continuity: A limit can exist even if the function is discontinuous. Continuity requires the limit to equal the function value.
4. Wrong standard limit application: lim(x→0)(sin x)/x = 1, but lim(x→0)(sin x) = 0 and lim(x→0) x = 0 separately. Don't confuse these.
5. Sign errors in piecewise functions: When computing LHL from the left, use the left branch formula. When computing RHL from the right, use the right branch formula.
6. Not rationalising when needed: For limits involving square roots, rationalise using the conjugate. For example, lim(x→0)(√(1+x) − 1)/x — multiply by (√(1+x) + 1)/(√(1+x) + 1).
10. Exam Checklist
- [ ] Can evaluate limits by direct substitution, factorisation, and rationalisation
- [ ] Know all standard limits (sin x/x, eˣ−1/x, ln(1+x)/x, etc.)
- [ ] Can use the squeeze theorem for limits like lim(x→0) x·sin(1/x)
- [ ] Can find LHL and RHL for piecewise functions
- [ ] Understand the three conditions for continuity (defined, limit exists, limit = value)
- [ ] Can classify discontinuities: removable, jump, infinite, oscillatory
- [ ] Know that polynomials are continuous everywhere
- [ ] Can handle limits involving trigonometric substitutions
- [ ] Remember lim(x→0)(1−cos x)/x² = 1/2 (a frequently tested identity)
- [ ] Can evaluate limits of the form lim(x→∞)(1 + a/x)^(bx) = e^(ab)