Home >> NCERT Solutions >> Class 12 >> Maths >> Chapter 5 – Continuity And Differentiability – Exercise 5.1
NCERT Solutions for Class 12 Maths Chapter 5 – Continuity and Differentiability Exercise 5.1
Quick Navigation – Jump to Question
Chapter 5 – Continuity and Differentiability Exercise 5.1
1.
Flashcard for Question 1
Quick Tip: Linear functions f(x) = ax + b are continuous everywhere; show lim_{x→c} f(x) = a·c + b = f(c) using limit laws – so continuity holds at x = 0, −3, and 5.
2.
Flashcard for Question 2
Quick Tip: For polynomials like f(x) = 2x² − 1, continuity holds everywhere. Just check lim_{x→3} f(x) = f(3) = 17, so the function is continuous at x = 3.
3.
4.
Flashcard for Question 4
Quick Tip: Use limit laws: limₓ→n xⁿ = (limₓ→n x)ⁿ = nⁿ = f(n), hence continuous at x = n.
Common Mistake: Expanding (xⁿ − nⁿ) unnecessarily instead of directly applying limit laws.
Exam Insight: In exams, a clean argument using limit continuity rules for polynomials is enough – no need for lengthy proofs.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
Flashcard for Question 19
Quick Tip: At integers, left-hand limit → 1, right-hand limit → 0, but g(n) = 0. Since LHL ≠ RHL, g(x) is discontinuous at all integers.
Common Mistake: Taking [x] as nearest integer instead of greatest integer ≤ x, which changes the limits.
Exam Insight: These “fractional part” functions are standard—exam expects you to compute LHL and RHL separately and show they’re unequal at integers.
20.
Flashcard for Question 20
Quick Tip: Since f(x) is a sum of continuous functions (x², −sin x, and constant 5), it is continuous everywhere. So at x = π, limₓ→π f(x) = f(π).
21.
22.
Flashcard for Question 22
Quick Tip: cos x is continuous for all real x. sec x, cosec x, and cot x are continuous wherever they’re defined (i.e., denominator ≠ 0).
Common Mistake: Forgetting to exclude points where sin x = 0 (for csc, cot) or cos x = 0 (for sec).
23.
25.
26.
27.
28.
29.
30.
31.
Flashcard for Question 31
Quick Tip: Composition of continuous functions is continuous: x² is continuous, cos x is continuous ⇒ cos(x²) is continuous everywhere.
Common Mistake: Trying to prove continuity at each point separately instead of using the composition rule.
32.
Flashcard for Question 32
Quick Tip: Since cos x is continuous and the modulus function |·| is continuous, their composition |cos x| is also continuous everywhere.
Common Mistake: Assuming |·| introduces discontinuity—remember it only changes the sign, not continuity.
33.
Flashcard for Question 33
Quick Tip: Both |x| and sin x are continuous, so their composition sin|x| is continuous everywhere (especially check at x = 0).
34.
Flashcard for Question 34
Quick Tip: Break into intervals around the points where expressions inside modulus change sign (x = 0 and x = –1). Check continuity at those points.
Common Mistake: Students often forget to test both points where the inside of modulus = 0, leading to missing discontinuities.
Exam Insight: Modulus-based piecewise functions are a favourite in exams; examiners expect interval-wise definition and checking limits at junctions.
Download Exercise 5.1 NCERT Solutions PDF
You can download the PDF from the link below for offline study
Class 12 Maths Chapter 5 – Continuity And Differentiability: All Exercises
| Exercise | Link |
|---|---|
| Exercise 5.2 | View Solutions |
| Exercise 5.3 | View Solutions |
| Exercise 5.4 | View Solutions |
| Exercise 5.5 | View Solutions |
| Exercise 5.6 | View Solutions |
| Exercise 5.7 | View Solutions |
| Miscellaneous Exercise | View Solutions |
Class 12 Continuity And Differentiability- Exercise 5.1 Overview
Welcome to a completely new chapter where your math knowledge advances a level! Examining limits again and then guiding you into the realm of continuity sets the tone in Class 12 NCERT Solutions Exercise 5.1. Differentiability It’s like preparing the basis before you start a major construction project. A fundamental concept in calculus and practical applications, this exercise clarifies whether a function behaves naturally or not.
The 2025 revised NCERT syllabus gives more weight on clear concept understanding. Exercise 5.1 helps you to clearly see what it means for a function to be continuous at a location and over an interval. By means of its methodical challenges, you will learn how to test continuity using left-hand and right-hand limits—a fundamental component of calculus logic.
The way Visual and approachable Continuity and Differentiability Class 12 NCERT Solutions Exercise 5.1 may be is one of its best aspects. Imagine driving a car; a smooth ride results from a route free of unexpected shocks or breaks. Continuity is all about exactly that! This exercise teaches you how to verify if a function has any “jumps,” or “breaks,” thereby strengthening that real-world relationship.
Whether you’re getting ready for your board tests or building the foundation for admission tests, you really must become proficient in this activity. It opens the road for all that follows: differentiability, the chain rule, and more. Therefore, slow down, grasp every step, and ensure that every idea in Continuity and Differentiability Class 12 NCERT Solutions Exercise 5.1 appeals to you.
FAQs – Continuity And Differentiability Class 12 Exercise 5.1 NCERT
You check if the left-hand limit, right-hand limit, and the value of the function at a point are all equal. If they are, the function is continuous at that point.
Start by focusing only on continuity in this exercise. Differentiability will come later. Remember, if a function is differentiable, it’s always continuous — but not the other way around.
You’ll often get 2 to 3 mark questions asking you to prove continuity at a point or over an interval — using basic limit definitions.
You can explore our step-by-step solutions to all questions from Continuity and Differentiability Class 12 NCERT Solutions Exercise 5.1 right here on Cogniks. Start practicing now and strengthen your concept clarity!
The best way is to revise with quick notes or flashcards after every session. Bookmark this page and come back to practice again before your next test!