NCERT Solutions for Class 11 Maths Chapter 13 – Limits and Derivatives
Subject: Mathematics | Class: 11 | Chapter: 13 | Board: CBSE | Curriculum: New NCERT
What Are NCERT Solutions for Class 11 Maths Chapter 13 – Limits and Derivatives?
Chapter 13 is where Class 11 Mathematics touches calculus for the first time. The word "calculus" often carries a reputation for difficulty, but its foundational idea — the limit — is actually one of the most natural concepts in mathematics. A limit asks: what value does a function approach as the input gets closer and closer to some point? That question, asked precisely, is the engine of all of calculus.
NCERT solutions introduces limits through examples and algebra rather than the formal epsilon-delta definition, making this chapter accessible for Class 11 students. From limits, the chapter moves to derivatives — the rate at which a function changes. The derivative of a function at a point is defined as a limit (the limit of the difference quotient), and then computed using rules that are derived from that limit definition. For all Chapters must, read NCERT Solutions for Class 11 Maths and subject-wise NCERT Solutions for Class 11.
This chapter is foundational for Class 12 Calculus (Continuity, Differentiability, Applications of Derivatives, Integration), so understanding it thoroughly now pays dividends for an entire year. Students who rush through this chapter and only memorise the standard derivative formulas without understanding where they come from consistently struggle in Class 12. The solutions here show both the first-principles method (using the limit definition) and the rule-based method, because CBSE sometimes specifically asks for the first-principles approach.
Download PDF – All Exercises of NCERT Solutions for Class 11 Maths Chapter 13 Limits and Derivative
| Exercise | Topic Covered | Number of Questions |
|---|---|---|
| Exercise 13.1 | Limits – Algebraic Evaluation, Standard Limits | 32 Questions |
| Exercise 13.2 | Derivatives – First Principles, Differentiation Rules | 11 Questions |
| Miscellaneous | Mixed Limits and Derivatives Problems | 30 Questions |
Chapter 13 – Limits and Derivatives: Concepts, Explanation and Key Tables
Understanding a Limit Intuitively
The limit of f(x) as x approaches a is the value f(x) gets arbitrarily close to (but not necessarily equals) as x gets arbitrarily close to a. We write: lim (x→a) f(x) = L.
A limit exists at x = a if and only if the left-hand limit equals the right-hand limit: lim (x→a⁻) f(x) = lim (x→a⁺) f(x)
Standard Limit Formulas
| Limit | Value | Notes |
|---|---|---|
| lim (x→a) [xⁿ – aⁿ] / (x – a) | naⁿ⁻¹ | Valid for all rational n; derived from factoring |
| lim (x→0) sin x / x | 1 | x must be in radians; most important trig limit |
| lim (x→0) tan x / x | 1 | Follows from sin x/x and cos x → 1 |
| lim (x→0) (1 – cos x) / x | 0 | |
| lim (x→0) (1 – cos x) / x² | 1/2 | |
| lim (x→0) (eˣ – 1) / x | 1 | |
| lim (x→0) (aˣ – 1) / x | log a | General exponential limit |
| lim (x→0) log(1 + x) / x | 1 | |
| lim (x→∞) (1 + 1/x)ˣ | e | Definition of Euler's number |
Algebra of Limits
If lim (x→a) f(x) = L and lim (x→a) g(x) = M, then:
| Operation | Result |
|---|---|
| lim [f(x) + g(x)] | L + M |
| lim [f(x) – g(x)] | L – M |
| lim [f(x) × g(x)] | L × M |
| lim [f(x) / g(x)] | L / M (provided M ≠ 0) |
| lim [k × f(x)] | k × L |
| lim [f(x)]ⁿ | Lⁿ |
Derivative — Definition and First Principles
The derivative of f(x) at x = a is defined as:
f′(a) = lim (h→0) [f(a+h) – f(a)] / h
This is the first-principles definition. For a general function f(x), the derivative function f′(x) (or df/dx) is:
f′(x) = lim (h→0) [f(x+h) – f(x)] / h
CBSE specifically asks to "differentiate from first principles" in certain questions — in those cases, the limit definition must be used, not the standard rules.
Standard Derivative Formulas
| Function f(x) | Derivative f′(x) | Condition |
|---|---|---|
| xⁿ | nxⁿ⁻¹ | Power rule; n is any real number |
| sin x | cos x | |
| cos x | –sin x | Note the negative sign |
| tan x | sec²x | |
| cot x | –cosec²x | |
| sec x | sec x tan x | |
| cosec x | –cosec x cot x | |
| eˣ | eˣ | |
| aˣ | aˣ log a | |
| log x | 1/x | x > 0 |
| constant | 0 |
Rules of Differentiation
| Rule | Expression | Formula |
|---|---|---|
| Sum Rule | d/dx [f + g] | f′ + g′ |
| Difference Rule | d/dx [f – g] | f′ – g′ |
| Product Rule | d/dx [f × g] | f′g + fg′ |
| Quotient Rule | d/dx [f/g] | (f′g – fg′) / g² |
| Constant Multiple | d/dx [k × f] | k × f′ |
| Chain Rule (introduced informally) | d/dx [f(g(x))] | f′(g(x)) × g′(x) |
Study Tips for Chapter 13
- Never substitute x = a directly into a limit if the result is 0/0 — this is the indeterminate form. Always factorise, rationalise, or use a standard limit formula to resolve it before substituting.
- For first-principles differentiation, set up [f(x+h) – f(x)]/h carefully, expand the numerator, cancel h, and only then take the limit h → 0.
- The product rule is easily misremembered as f′ × g′ (wrong). It is f′g + fg′ — two terms, not one.
- Practise the quotient rule on fractions involving trig functions — these are the most common miscellaneous exercise problems in this chapter.
FAQs on NCERT Solutions Class 11 Maths Chapter 13 Limits and Derivatives
Limits are mathematical concepts that describe the value a function approaches as the input approaches a particular point. They form the foundation of calculus and help students understand the behavior of functions near specific values. In Chapter 13, students learn methods for evaluating limits and solving problems involving algebraic expressions. NCERT Solutions explain these concepts with detailed examples and step-by-step procedures. Understanding limits is important because they are widely used in advanced mathematics, physics, economics, and engineering. A strong grasp of limits helps students prepare for higher studies and develop analytical problem-solving skills.
A derivative measures the rate at which one quantity changes with respect to another. It is one of the most important concepts in calculus and has applications in science, engineering, economics, and business. In Class 11, students are introduced to the basic idea of derivatives and their geometric interpretation. NCERT Solutions explain derivative concepts in a simple and structured manner, making them easier to understand. Learning derivatives helps students analyze changes, solve optimization problems, and prepare for advanced calculus topics that are covered extensively in Class 12 Mathematics.
Limits and derivatives are closely related because derivatives are defined using limits. To find the derivative of a function, students calculate the limit of a specific difference quotient as the change in the variable approaches zero. Understanding limits is therefore essential for understanding derivatives. NCERT Solutions demonstrate this relationship through solved examples and detailed explanations. By learning how limits lead to derivatives, students gain a deeper understanding of calculus and develop the mathematical reasoning needed for higher-level topics and competitive examinations.
NCERT Solutions provide detailed and accurate explanations for every exercise in the chapter. They help students understand fundamental concepts, formulas, and problem-solving techniques through step-by-step methods. The solutions simplify complex questions and improve conceptual clarity. Students can use them for homework, revision, and examination preparation. Regular practice enhances logical thinking, analytical skills, and confidence. Since limits and derivatives are the building blocks of calculus, mastering NCERT Solutions helps students establish a strong foundation for Class 12 Mathematics and competitive entrance examinations.