JEE Main 2025: Limit Continuity and Differentiability JEE Main Notes

The  Limit Continuity and Differentiability chapter in calculus covers the important feature of understanding how functions behave as they approach certain points or values. It delves into the concepts of limits, continuity, and differentiability, which are foundational for understanding the behaviour and properties of functions in calculus and real analysis. This is one of the important chapters for JEE Main Exams. Have a look at the number of questions asked from the Limit Continuity and Differentiability chapter in the past five years of JEE Main exam.

You can find that 3 questions have been asked from the past five years of JEE main exam. The teachers worked hard to ensure the Limit Continuity and Differentiability notes perfectly matched the JEE Main syllabus. So, gear up to succeed – grab your Limit Continuity and Differentiability Notes now and shine in JEE Main 2025!

Top Concepts to Study in Limit Continuity and Differentiability for JEE Main 2025

Here you will look at the main topics you should go over in Limit Continuity and Differentiability. These crucial topics can truly improve your preparation. Good at them will give you a big advantage in the upcoming Test. So, let's discuss the essential topics you need to consider for success in JEE Main 2025.

• Limits of Functions

• Continuity and Discontinuity

• Differentiability and Derivatives

• Differentiation Rules

• Applications of Derivatives

• Mean Value Theorem

• Rolle's Theorem

• L'Hopital's Rule

• Continuity of Composite Functions

• Differentiability of Composite Functions

Important Formulas in Mathematics Limit, Continuity and Differentiability JEE Main Notes PDF

Here are some essential formulas for the Limit, Continuity, and Differentiability topics in JEE Main Mathematics:

Limits

1. Basic Limit Property: $\lim_{{x \to a}} f(x) = f(a)$ (for continuous functions at $x = a$).

2. Limits at Infinity:

   - $\lim_{{x \to \infty}} \dfrac{1}{x} = 0$

   - $\lim_{{x \to 0}} \dfrac{1}{x} = \infty$ or undefined.

3. Important Limits:

   - $\lim_{{x \to 0}} \dfrac{\sin x}{x} = 1$

   - $\lim_{{x \to 0}} \dfrac{\tan x}{x} = 1$

   - $\lim_{{x \to 0}} \dfrac{1 - \cos x}{x^2} = \dfrac{1}{2}$

   - $\lim_{{x \to \infty}} \left(1 + \dfrac{1}{x}\right)^x = e$

4. L'Hôpital's Rule: If $\lim_{{x \to a}} \dfrac{f(x)}{g(x)}$ results in $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$, then

\[\lim_{{x \to a}} \dfrac{f(x)}{g(x)} = \lim_{{x \to a}} \dfrac{f'(x)}{g'(x)}\]

Continuity

1. Condition for Continuity at $x = a$:

\[\lim_{{x \to a^-}} f(x) = \lim_{{x \to a^+}} f(x) = f(a)\]

Differentiability

1. Basic Differentiation Rules:

   - Power Rule: $\dfrac{d}{dx} (x^n) = n x^{n-1}$

   - Product Rule: $\dfrac{d}{dx} (uv) = u'v + uv'$

   - Quotient Rule: $\dfrac{d}{dx} \left(\dfrac{u}{v}\right) = \dfrac{u'v - uv'}{v^2}$

   - Chain Rule: If $y = f(g(x))$, then $\dfrac{dy}{dx} = f'(g(x)) \cdot g'(x)$

2. Trigonometric Functions:

   - $\dfrac{d}{dx} (\sin x) = \cos x$

   - $\dfrac{d}{dx} (\cos x) = -\sin x$

   - $\dfrac{d}{dx} (\tan x) = \sec^2 x$

3. Exponential and Logarithmic Functions:

   - $\dfrac{d}{dx} (e^x) = e^x$

   - $\dfrac{d}{dx} (\ln x) = \dfrac{1}{x}$

4. Implicit Differentiation: Differentiate both sides of an equation with respect to $x$, treating $y$ as a function of $x$.

Higher Order Derivatives

1. Leibniz Rule for nth Derivative:

\[\dfrac{d^n}{dx^n} (uv) = \sum_{{k=0}}^{n} \binom{n}{k} \cdot u^{(k)} v^{(n-k)}\]

Mean Value Theorem (MVT)

If a function $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists $c \in (a, b)$ such that:

\[f'(c) = \dfrac{f(b) - f(a)}{b - a}\]

The Significance of Limit Continuity and Differentiability Notes for JEE Main 2025

Dive into the exhilarating world of 'Limit, Continuity, and Differentiability' – a chapter that unlocks the secrets of functions and their behaviour like never before. Whether you're a curious learner or a future mathematician, get ready to embark on a journey that reveals the captivating interplay between mathematical concepts and real-world applications through our JEE Mathematics Limit, Continuity and Differentiability Notes

Limit: Peeking at Infinity and Beyond!

• Imagine pushing numbers to their limits! Discover the essence of limits, where numbers approach certain values as if edging closer to infinity.

• Engage in interactive examples that illustrate how limits redefine a function's behaviour at specific points.

• Grasp the concept of one-sided limits, where numbers tiptoe from the left or the right, shaping the character of a function.

Continuity: Bridging Gaps with Mathematical Harmony

• Picture functions without abrupt jumps or breaks – that's continuity! Uncover how functions can smoothly transition from one point to another.

• Delve into the world of continuous functions, exploring their properties and the elegance they bring to mathematical landscapes.

• Explore the Intermediate Value Theorem, a thrilling ride that assures you there's always a number to hold the hand of another number in between!

Differentiability: The Art of Tangents and Derivatives

• Ever wondered how a curve cuddles up with a tangent line? Differentiability unveils the magic behind tangents and derivatives.

• Traverse through differentiation rules and techniques, uncovering the secrets of finding slopes, instantaneous rates of change, and much more.

• Immerse yourself in real-world applications – from tracking changing velocities to understanding how things grow and decay.

Mastering the Art of Limits, Continuity, and Differentiability for JEE Main: your Blueprint for Success

• Break down complex concepts with interactive examples, guided exercises, and relatable scenarios.

• Level up your problem-solving prowess by conquering challenges that strengthen your analytical skills.

• Discover the connections between these concepts and other mathematical domains, paving the way for holistic understanding.

Prepare to unravel the mysteries of Limit, Continuity, and Differentiability as you navigate through this interactive journey. Embrace the excitement of mathematical exploration, and watch as these concepts transform from abstract notions into powerful tools in your JEE Main arsenal. Are you ready to witness the captivating dance of numbers and functions? Let the adventure begin!

Strategic Approach: Maximising Learning with Formulas and Revision Notes

When you want to learn effectively, you can use a smart strategy by combining formulas and revision notes. These two things work together nicely. Formulas are like short explanations of important ideas. When you use them with detailed revision notes, you get a complete way to learn. Revision notes give you a strong base, and formulas help you understand better and solve problems. This mix makes hard things easier and helps you remember better. By using this strategy, you can improve how well you understand and do well in your studies.

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Maths Notes: Chapter-wise links 2025

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Further Resources for JEE Main 2025 -  Limit Continuity and Differentiability

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Explore Further: Supplementary Resources for JEE Main 2025 Limit Continuity and Differentiability

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Use these tools to improve comprehension and test-taking skills, boosting your JEE Main 2025 performance!

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In your journey to conquer JEE Main 2025, Limit Continuity and Differentiability Notes are your trusted guide. They simplify complex concepts, highlight key points, and provide practice, making your preparation effective. With Vedantu's JEE Main notes, you will understand better, solve problems confidently, and excel in the exam. Download Limit Continuity and Differentiability Revision Notes for FREE to sharpen your skills and triumph in JEE Main 2025.