5.1|Plus two Mathematics |Continuity and Differentiability 5.1| Limit & Continuity 5.1|+2 Maths|KEAM

Introduction

This tutorial provides a step-by-step guide to understanding the concepts of continuity and differentiability in mathematics, specifically tailored for Plus Two students preparing for exams like KEAM, JEE, and IISER. By following these steps, you will gain a clear understanding of limits, continuity, and differentiation.

Step 1: Understanding Limits

• Definition of Limits: A limit is a value that a function approaches as the input approaches a certain point.
• Notation: The limit of a function f(x) as x approaches a is denoted as:

  lim (x -> a) f(x)
  

• Calculating Limits:
  • Identify the point at which you want to find the limit.
  • Substitute the value into the function if possible.
  • If direct substitution results in an indeterminate form (like 0/0), use algebraic manipulation or L'Hôpital's Rule.

Practical Tip

• Always check if the function can be simplified before applying L'Hôpital's Rule.

Step 2: Exploring Continuity

• Definition of Continuity: A function is continuous at a point if:

  1. The function is defined at that point.
  2. The limit exists at that point.
  3. The limit equals the function's value at that point.

• Types of Discontinuity:

  • Removable: A hole in the graph, which can be fixed.
  • Jump: Sudden changes in function values.
  • Infinite: The function approaches infinity.

Practical Tip

• Graphing the function can help visualize continuity and identify types of discontinuities.

Step 3: Differentiability and Its Relationship with Continuity

• Definition of Differentiability: A function is differentiable at a point if it has a defined derivative at that point.
• Connection to Continuity: A function must be continuous at a point to be differentiable there, but continuity alone does not guarantee differentiability.

Key Points

• If a function has a sharp corner or cusp, it may be continuous but not differentiable.
• The derivative represents the slope of the tangent line to the function at a certain point.

Step 4: Finding Derivatives

• Basic Derivative Rules:
  • Power Rule: If f(x) = x^n, then f'(x) = n*x^(n-1).
  • Product Rule: If f(x) = u(x)*v(x), then f'(x) = u'v + uv'.
  • Quotient Rule: If f(x) = u/v, then f'(x) = (u'v - uv')/v^2.

Example

For f(x) = x^3 - 3x + 2:

• To find f'(x):

  f'(x) = 3x^2 - 3
  

Step 5: Practice Problems

• Work through previous year exam questions related to limits, continuity, and differentiability.
• Attempt NCERT exercise problems to reinforce your understanding.

Conclusion

In this tutorial, we covered the fundamentals of limits, continuity, and differentiability which are essential for mastering Plus Two mathematics. Understanding these concepts is crucial for success in competitive exams like KEAM and JEE. As a next step, consider practicing with additional problems and reviewing any areas where you feel less confident.