Limits and Derivatives — Class 11 MCQs with Answers
Class 11 CBSE Mathematics · Chapter 13
90 practice questions · 30 Easy · 30 Medium · 30 Hard · Updated
Practise the most important Class 11 CBSE Mathematics questions from Chapter 13, "Limits and Derivatives". You get 9 timed quizzes made from 90 NCERT-based MCQs, with answers and explanations. The questions are split into 30 Easy, 30 Medium and 30 Hard. Warm up on the basics, then move on to the exam-level questions that set top scorers in CBSE & Maharashtra HSC Board exams, JEE Main, MHT-CET and JEE Advanced apart.
To score well in "Limits and Derivatives", focus on fast problem-solving, formula recall and step-by-step working. Each MCQ here is timed and uses exam-style marking (+4 correct, −1 wrong, 0 skipped). This trains you to stay accurate under time pressure, as real papers need. Every question has a short explanation, so a wrong answer becomes a quick lesson. It is the fastest way to fix gaps before a test.
Use this chapter for focused revision. Start with the Easy set to check your basics on Limits and Derivatives, then move to Medium and Hard to practise applying them. Your accuracy, streaks and XP save automatically. This chapter also adds to your overall Class 11 Mathematics mastery score. 14 sample questions are solved in full below, with the answer and a worked explanation. Sign in free to start practising.
Key concepts: Limits and Derivatives (Class 11 Mathematics)
A limit captures the value a function heads toward as its input nears a point, even where the function is undefined. This chapter develops the algebra of limits, the standard trigonometric, exponential and algebraic limits, and the derivative as a limit — the instantaneous rate of change — with rules for fast differentiation.
- Intuitive limit
- lim(x→a) f(x) = L means f(x) can be made as close to L as we like by taking x close enough to a, whether or not f(a) itself exists.
- One-sided limits
- The left-hand limit lets x approach a from below and the right-hand limit from above; each describes the function's behaviour on one side only.
- Existence of a limit
- A limit exists only when the left- and right-hand limits are both finite and equal; a jump between them means the limit fails to exist.
- Algebra of limits
- The limit of a sum, difference, product or quotient is that same operation on the individual limits, provided each exists (and the denominator ≠ 0).
Limits and Derivatives — important questions & MCQs with answers (Class 11 Mathematics)
14 solved questions from this chapter's difficulty levels, each with its answer and explanation. The other 76 are timed and scored when you sign in.
- Q1Easy
lim(x→2) (x²) =
A.∞B.2C.0D.4✓ CorrectAnswer: D. 4
Explanation: For a continuous function such as x², the limit at a point equals the value there: substitute x = 2 to get 2² = 4. Writing just x = 2 as the answer confuses the input with the limit.
- Q2Easy
d/dx (xⁿ) =
A.xⁿB.n·xⁿ⁻¹✓ CorrectC.nxD.xⁿ⁺¹/(n+1)Answer: B. n·xⁿ⁻¹
Explanation: Power rule for derivatives: bring the exponent down as a coefficient and reduce it by one, giving n·xⁿ⁻¹. Leaving xⁿ unchanged or forgetting to drop the exponent is the usual slip.
- Q3Easy
lim (x→a) f(x) = L means:
A.f(x) approaches L as x approaches a✓ CorrectB.f(a) = LC.f(x) = L for all xD.L is the maximumAnswer: A. f(x) approaches L as x approaches a
Explanation: A limit describes what f(x) approaches as x nears a — the value L, not necessarily the value at that point itself. f need not even be defined there for the limit to exist.
- Q4Easy
Derivative of f(x) at x = a is:
A.lim (h→0) [f(a+h) − f(a)]/h✓ CorrectB.f(a)C.f'(a) onlyD.f(a)/aAnswer: A. lim (h→0) [f(a+h) − f(a)]/h
Explanation: By definition, the derivative of f at a point is the limit of the difference quotient, lim(h→0) [f(x+h) − f(x)]/h, evaluated at that point as h → 0. Using a fixed nonzero h instead of taking the limit misses the essential step.
- Q5Easy
Derivative of constant c is:
A.0✓ CorrectB.cC.1D.c · xAnswer: A. 0
Explanation: A constant never changes as x varies, so its derivative is 0 — there's no rate of change to measure. Writing the constant itself as the derivative instead of 0 is the usual error.
- Q6Easy
Derivative of x³ is:
A.3x²✓ CorrectB.x²C.3xD.x³/3Answer: A. 3x²
Explanation: Power rule: d/dx(x³) = 3x², bring the exponent 3 down as the coefficient and reduce it by one. Leaving the exponent at 3, or forgetting the coefficient 3, is the common slip.
- Q7Medium
lim(x→0) (sin x)/x =
A.∞B.0C.1✓ CorrectD.undefinedAnswer: C. 1
Explanation: Standard trig limit: (sin x)/x → 1 as x → 0, valid only when x is measured in radians. Treating this 0/0-looking form as indeterminate and guessing 0 is the common trap.
- Q8Medium
d/dx (sin x) =
A.tan xB.−cos xC.−sin xD.cos x✓ CorrectAnswer: D. cos x
Explanation: The derivative of sin x is cos x, one of the standard trig derivatives to memorize. Mixing it up with the derivative of cos x, which is −sin x, is the common confusion.
- Q9Medium
lim (x→2) (x² − 4)/(x − 2) =
A.4✓ CorrectB.0C.2D.UndefinedAnswer: A. 4
Explanation: Direct substitution gives 0/0, so factor: (x² − 4)/(x − 2) = (x + 2)(x − 2)/(x − 2) = x + 2, which → 4 as x → 2. Stopping at the 0/0 form and calling the limit undefined is the trap.
- Q10Medium
The value of the derivative of x² sin x at x = 0 is:
A.−1B.1C.2D.0✓ CorrectAnswer: D. 0
Explanation: The product rule gives 2x sin x + x² cos x, which vanishes at 0.
- Q11Medium
Derivative of 3x⁴ − 2x² + 5x is:
A.x³ − 2x + 5B.12x³ − 4xC.12x³ − 4x + 5✓ CorrectD.12x³ + 4xAnswer: C. 12x³ − 4x + 5
Explanation: Differentiate term by term with the power rule: d/dx(3x⁴) = 12x³, d/dx(−2x²) = −4x, d/dx(5x) = 5. Dropping the constant +5 from the linear term is the common slip.
- Q12Hard
lim(x→0) (1−cos x)/x² =
A.1B.0C.1/2✓ CorrectD.∞Answer: C. 1/2
Explanation: Standard limit: (1 − cos x)/x² → 1/2 as x → 0, since 1 − cos x ≈ x²/2 for small x. Confusing it with the simpler (1 − cos x)/x → 0, a different limit, is the common trap.
- Q13Hard
d/dx (uv) =
A.u'v − uv'B.u'v'C.(uv)'D.u'v + uv'✓ CorrectAnswer: D. u'v + uv'
Explanation: Product rule: d/dx(uv) = u′v + uv′ — differentiate each factor in turn and add the two results. Dropping one term, or multiplying the derivatives together as u′v′, is the usual error.
- Q14Hard
lim (x→0) (a^x − 1)/x =
A.ln a✓ CorrectB.aC.1D.0Answer: A. ln a
Explanation: Standard limit: (aˣ − 1)/x → ln a as x → 0, generalizing (eˣ − 1)/x → 1, the special case a = e where ln e = 1. Forgetting the ln and answering just a is the common trap.
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Start this chapter free →Limits and Derivatives — FAQs
What are the key concepts in Class 11 Mathematics Limits and Derivatives?+
A limit captures the value a function heads toward as its input nears a point, even where the function is undefined. This chapter develops the algebra of limits, the standard trigonometric, exponential and algebraic limits, and the derivative as a limit — the instantaneous rate of change — with rules for fast differentiation. Key ideas include Intuitive limit, One-sided limits, Existence of a limit, Algebra of limits.
What does Class 11 Mathematics Chapter 13 (Limits and Derivatives) cover on XamBaaz?+
It has 90 NCERT-based MCQs on "Limits and Derivatives": 30 Easy, 30 Medium and 30 Hard. Together they make 9 timed quizzes, and you never get the same set twice. Every question has an instant explanation. They help you prepare for CBSE & Maharashtra HSC Board exams, JEE Main, MHT-CET and JEE Advanced.
Are these "Limits and Derivatives" questions free to practise?+
Yes. Sign in with Google to practise "Limits and Derivatives" free. Full unlimited access is ₹999/year on a launch offer until 1 December 2026. No chapter is charged separately.
How should I revise "Limits and Derivatives" for the exam?+
Start with the Easy quiz to check your basics, then try Medium and Hard to practise applying them. There are 9 timed quizzes on this chapter, so you can come back for a fresh set instead of one you have seen. Read each explanation, retry the questions you miss, and track your accuracy until it stays high.
Are these "Limits and Derivatives" MCQs available with answers?+
Yes. 14 sample questions are shown here in full, each with the correct option and a step-by-step "Why" explanation. Sign in free with Google to start practising, with instant scoring.
Is there negative marking in the "Limits and Derivatives" quizzes?+
Yes. The timed quizzes use exam-style marking: +4 for a right answer, −1 for a wrong one and 0 for a skip, the same negative marking as JEE Main and JEE Advanced. MHT-CET and CBSE board papers have no negative marking. Our mocks for those are scored their way.
What are the important questions from Limits and Derivatives (Class 11 Mathematics)?+
The questions that matter most test Intuitive limit, One-sided limits, Existence of a limit, Algebra of limits. This page shows 14 solved important MCQs with answers and explanations; all 90 questions on the chapter are available as timed quizzes once you sign in.
Is there an online quiz for Limits and Derivatives?+
Yes — Class 11 Mathematics Limits and Derivatives has timed online quizzes at Easy, Medium and Hard levels, with instant scoring and a worked explanation on every question. The first quiz on the chapter is free.