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.