Top Strategies to Master Limit Continuity and Differentiability for JEE Main Mock Tests
Limit, Continuity, and Differentiability are fundamental concepts in JEE Mathematics, laying the groundwork for calculus and advanced mathematical reasoning. Mastering these topics not only boosts your problem-solving speed but also ensures seamless progression in solving real exam questions on rate of change, tangent behavior, and function properties. Challenge yourself with this chapter-specific mock test to sharpen your skills and ace the JEE Main!
Mock Test Instructions for the Limit, Continuity and Differentiability:
- 20 questions from Limit, Continuity and Differentiability
- Time limit: 20 minutes
- Single correct answer per question
- Correct answers appear in bold green after submission
How Do JEE Mock Tests Help in Mastering Limits, Continuity, and Differentiability?
- Mock tests for Limit, Continuity, and Differentiability help in quickly identifying your calculation mistakes with limits and function behaviors.
- Timed online tests improve your ability to handle trick questions on left-hand and right-hand limits, especially in exam conditions.
- Regular practice uncovers weak points in differentiability as well as understanding of discontinuities at given points.
- Mock tests enable mastery over techniques like L'Hospital's Rule, standard limits, and piece-wise function continuity.
- Utilizing detailed test feedback helps focus your revision on important JEE formulas and commonly tested concepts.
Sharpen Your Calculus Foundation with Limit, Continuity, and Differentiability JEE Mock Tests
- Targeted mock tests simulate real exam pressure, boosting your time management in lengthy calculus questions.
- Expertly designed problems help in visualizing graphs for continuity and checking differentiability at critical points.
- Learn to tackle multiple-choice problems on algebraic and trigonometric limits using smart shortcuts revealed in test practice.
- Consistent mock test practice makes you proficient with limit properties, removable/discontinuity types, and the definition of derivatives.
- Immediate result analysis after mock tests helps track your progress and refines your approach for the actual JEE exam.
Subject-Wise Excellence: JEE Main Mock Test Links
| S.No. | Subject-Specific JEE Main Online Mock Tests |
|---|---|
| 1 | Online FREE Mock Test for JEE Main Chemistry |
| 2 | Online FREE Mock Test for JEE Main Maths |
| 3 | Online FREE Mock Test for JEE Main Physics |
Important Study Materials Links for JEE Exams
FAQs on Ace JEE Main 2025-26 with Mock Tests on Limit Continuity and Differentiability
1. What is a limit in calculus?
Limit is a fundamental concept in calculus that describes the value a function approaches as the input approaches a certain point. Limits are essential for defining continuity and differentiability of functions. For example, the limit of f(x) as x approaches a is denoted as limx→a f(x).
2. Explain continuity at a point with an example.
A function is continuous at a point if the following three conditions are satisfied: (1) the function is defined at the point, (2) the limit of the function exists at that point, and (3) the value of the function at the point equals the limit. For example, f(x) = 2x is continuous everywhere because for any point a, limx→a f(x) = f(a).
3. How do you test the continuity of a function at x = a?
To test continuity at x = a:
• Check that f(a) is defined.
• Find the left-hand limit (LHL) and right-hand limit (RHL) as x approaches a.
• Confirm LHL = RHL = f(a).
If all these hold, the function is continuous at x = a.
4. What are the types of discontinuities in a function?
Types of discontinuities include:
• Removable discontinuity – when the limit exists but is not equal to the function's value.
• Jump discontinuity – when LHL and RHL exist but are unequal.
• Infinite discontinuity – when the limit approaches infinity.
These occur at specific points in a function's domain.
5. State the formal definition of derivative (differentiability) at a point.
Differentiability at a point is formally defined as the existence of the following limit:
f'(a) = limh→0 [f(a+h) - f(a)] / h.
The function must be continuous at a and this limit must exist for differentiability.
6. If a function is differentiable at a point, is it continuous there?
Yes, if a function is differentiable at a point, it is always continuous at that point. However, the converse is not always true; a function may be continuous but not differentiable (for example, |x| at x = 0).
7. How do you evaluate basic limits using algebraic methods?
To evaluate basic limits algebraically:
• Simplify the expression, factorise if possible.
• Cancel common factors.
• Substitute the limiting value.
If direct substitution leads to 0/0, apply L'Hospital's Rule or rationalisation as per syllabus parity.
8. What is the limit of (sin x)/x as x approaches 0?
The standard result is limx→0 (sin x)/x = 1. This special limit is often used for trigonometric functions and derivatives.
9. Which functions are always continuous everywhere?
Polynomial functions, exponential functions, and trigonometric functions like sin x and cos x are always continuous for all real values of x. Rational functions are continuous where the denominator is non-zero.
10. What is meant by one-sided limits? Why are they important for continuity?
One-sided limits refer to the values that a function approaches from the left (LHL) or right (RHL) of a particular point. A function is continuous at a point only if both one-sided limits exist and are equal to each other and to the value of the function at that point.
11. How are limits, continuity, and differentiability related?
There is a hierarchy among these concepts:
• If a function is differentiable at a point, it is necessarily continuous there.
• If it is continuous, the limit at that point exists.
• The reverse does not always hold; a function can have a limit without being continuous or differentiable.
12. Can you give an example where a function is continuous but not differentiable?
Yes, the function f(x) = |x| is continuous at x = 0 but not differentiable at x = 0 because the left and right derivatives are not equal at that point. This forms a sharp corner on its graph.
