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Example 16 - Chapter 5 Class 12 Continuity and Differentiability
Last updated at Dec. 16, 2024 by Teachoo
Checking continuity at any point
Chapter 5 Class 12 Continuity and Differentiability
Concept wise
- Checking continuity at a given point
-
Checking continuity at any point
- Checking continuity using LHL and RHL
- Algebra of continous functions
- Continuity of composite functions
- Checking if funciton is differentiable
- Finding derivative of a function by chain rule
- Finding derivative of Implicit functions
- Finding derivative of Inverse trigonometric functions
- Finding derivative of Exponential & logarithm functions
- Logarithmic Differentiation - Type 1
- Logarithmic Differentiation - Type 2
- Derivatives in parametric form
- Finding second order derivatives - Normal form
- Finding second order derivatives- Implicit form
- Proofs
- Verify Rolles theorem
- Verify Mean Value Theorem
Transcript
Example 16 Prove that every rational function is continuous.Every Rational function 𝑓(𝑥) is of the form 𝒇(𝒙)= 𝒑(𝒙)/𝒒(𝒙) where 𝑞(𝑥) ≠ 0 & 𝑝, 𝑞 are polynomial functions Since 𝑝(𝑥) & 𝑞(𝑥) are polynomials, and we know that every polynomial function is continuous. Therefore, 𝒑(𝒙) & 𝒒(𝒙) both continuous By Algebra of continuous function If 𝑝, 𝑞 are continuous , then 𝒑/𝒒 is continuous. Thus, Rational Function 𝑓(𝑥)= 𝑝(𝑥)/𝑞(𝑥) is continuous for all real numbers except at points where 𝒒(𝒙) = 0
