Key Concepts and Practice Questions on Limit Continuity and Differentiability
Limit, Continuity, and Differentiability is a foundational chapter in JEE Maths, testing your understanding of calculus fundamentals such as standard limits, types of discontinuity, and various differentiation techniques. Strong command over these concepts is essential for both JEE Main and Advanced as they form the basis for many complex problems. Take this mock test to solidify your skills and gain the confidence to tackle any question from this crucial chapter!
Mock Test Instructions for the Limit Continuity And Differentiability Mock Test 1:
- 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 Can JEE Mock Tests Help You Master Limits, Continuity, and Differentiability?
- Identify recurring pitfalls in limits, discontinuity, and differentiability using time-bound mock tests.
- Use mock tests to strengthen speed and accuracy in evaluating left-hand and right-hand limits.
- Prepare for complex limit problems that appear in JEE by repeated targeted practice.
- Pinpoint weaknesses in differentiability concepts such as corner points and modulus functions through instant feedback.
- Track your chapter-wise progress, enabling a focused revision before the JEE exam.
Build Strong Calculus Foundation for JEE with Expert-Designed Mock Tests on Limit, Continuity & Differentiability
- Practice all critical topics: standard limits, types of discontinuity, L'Hospital's Rule, and differentiability checks.
- Expert-curated MCQs mimic latest JEE Main/Advanced paper pattern for the Limits & Continuity chapter.
- Immediate explanations post-submission clarify concepts and improve retention.
- Fine-tune your exam approach by managing time pressure in real test-like scenario.
- Boost overall calculus accuracy for the entire JEE Maths syllabus by mastering this foundation chapter.
Subject-Wise Excellence: JEE Main Mock Test Links
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| 1 | Online FREE Mock Test for JEE Main Chemistry |
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FAQs on Limit, Continuity, and Differentiability Mock Test for 2025
1. What is a limit in calculus?
A limit in calculus refers to the value that a function or expression approaches as the input (or variable) approaches a particular point. Limits help in understanding the behavior of functions near specific values and are fundamental to the study of continuity and differentiability.
2. How do you determine if a function is continuous at a point?
A function is continuous at a point x = a if the following three conditions are satisfied:
1. f(a) is defined.
2. The limit of the function as x approaches a exists.
3. The value of the function at a is equal to its limit as x approaches a.
In symbols: limx→a f(x) = f(a).
3. State the formal definition of a limit.
The formal or epsilon-delta definition states that the limit of f(x) as x approaches a is L if for every number ε > 0, there exists a number δ > 0 such that whenever 0 < |x - a| < δ, it follows that |f(x) - L| < ε.
4. What are the three conditions for differentiability at a point?
A function is differentiable at a point x = a if:
1. The function is defined at a.
2. The function is continuous at a.
3. The left-hand derivative and right-hand derivative at a exist and are equal.
5. Explain the difference between one-sided and two-sided limits.
A one-sided limit refers to the value a function approaches as the input approaches a point from one direction only (from the left or right). A two-sided limit considers the value from both directions. In order for the two-sided limit to exist, both one-sided limits must exist and be equal at that point.
6. What is the tangent line to a function at a given point?
The tangent line to a function at a given point is a straight line that just touches the curve at that point and has the same slope as the function at that point. The slope of this tangent is given by the derivative of the function at the specific point.
7. Can a function be continuous but not differentiable? Give an example.
Yes, a function can be continuous at a point but not differentiable there. For example, f(x) = |x| is continuous everywhere but not differentiable at x = 0 due to a sharp corner at that point.
8. What is the importance of limits in calculus?
Limits form the foundation of calculus as they are essential in defining derivatives and integrals. They help in analyzing function behavior near specific points and are crucial for solving problems involving continuity and differentiability.
9. How do you test continuity using limits in a piecewise function?
To check continuity in a piecewise function at a specific point, ensure that the left-hand limit (LHL), right-hand limit (RHL), and the function's value at that point are all equal. If LHL = RHL = f(a), the function is continuous at x = a.
10. What are some common techniques to evaluate limits?
Common techniques to find limits include:
• Direct substitution
• Factoring and cancelling common terms
• Rationalizing the numerator or denominator
• Using L'Hospital's Rule when limits yield indeterminate forms
11. What is the relation between continuity and differentiability?
A function must be continuous at a point to be differentiable there, but not all continuous functions are differentiable. Differentiability implies continuity, but continuity does not guarantee differentiability.
12. What are indeterminate forms and how are they resolved in limits?
Indeterminate forms in limits (such as 0/0 or ∞/∞) are expressions where the limit cannot be determined directly. These forms are resolved using algebraic simplification, L'Hospital's Rule, or by manipulating the expression to remove the indeterminacy.
