# Ex 5.1, 13

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

Ex 5.1, 13 Is the function defined by ๐ ๐ฅ๏ทฏ= ๐ฅ+5, ๐๐ ๐ฅโค1๏ทฎ& ๐ฅโ5 , ๐๐ ๐ฅ>1๏ทฏ๏ทฏ a continuous function? Given function is , ๐ ๐ฅ๏ทฏ= ๐ฅ+5, ๐๐ ๐ฅโค1๏ทฎ& ๐ฅโ5 , ๐๐ ๐ฅ>1๏ทฏ๏ทฏ Case 1 At x = 1 f is continuous at x = 1 if L.H.L = R.H.L = ๐ 1๏ทฏ i.e. if lim๏ทฎxโ 1๏ทฎโ๏ทฏ๏ทฏ ๐ ๐ฅ๏ทฏ = lim๏ทฎxโ 1๏ทฎ+๏ทฏ๏ทฏ ๐ ๐ฅ๏ทฏ = ๐ 1๏ทฏ Thus, L.H.L โ R.H.L โ f is discontinuous at ๐ =๐ Case 2 Let x = c , where c < 1 ๐ ๐ฅ๏ทฏ=๐ฅ+5 f is continuous at x = c if if lim๏ทฎxโ๐๏ทฏ ๐ ๐ฅ๏ทฏ=๐(๐) Thus, lim๏ทฎxโ๐๏ทฏ ๐ ๐ฅ๏ทฏ=๐ ๐๏ทฏ โ f is continuous for ๐ฅ =๐ less than 1. โ f is at continuous for all real numbers less than 1. Case 3 Let x = c (where c > 1) ๐ ๐ฅ๏ทฏ= ๐ฅโ5 f is continuous at x = c if lim๏ทฎxโ๐๏ทฏ ๐ ๐ฅ๏ทฏ=๐(๐) Thus lim๏ทฎxโ๐๏ทฏ ๐ ๐ฅ๏ทฏ=๐(๐) โ f is continuous for ๐ฅ =๐ ( c is greater than 1) โ f is continuous at all real numbers greater than 1. Hence, only x = 1 is point of discontinuity. โ f is continuous at all real point. Thus, f is continuous for all ๐โ๐โ{๐}.

Ex 5.1, 9
Important

Ex 5.1, 13 Important You are here

Ex 5.1, 16 Important

Ex 5.1, 18 Important

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Ex 5.1, 30 Important

Ex 5.1, 34 Important

Ex 5.2, 5 Important

Ex 5.2, 9 Important

Ex 5.2, 10 Important

Ex 5.3, 10 Important

Ex 5.3, 14 Important

Example 32 Important

Example 33 Important

Ex 5.5,6 Important

Ex 5.5, 7 Important

Ex 5.5, 11 Important

Ex 5.5, 16 Important

Ex 5.6, 7 Important

Ex 5.6, 11 Important

Example 41 Important

Ex 5.7, 14 Important

Example 42 Important

Ex 5.8, 5 Important

Example 44 Important

Example 45 Important

Example 47 Important

Misc 6 Important

Misc 15 Important

Misc 16 Important

Misc 23 Important

Class 12

Important Question for exams Class 12

- Chapter 1 Class 12 Relation and Functions
- Chapter 2 Class 12 Inverse Trigonometric Functions
- Chapter 3 Class 12 Matrices
- Chapter 4 Class 12 Determinants
- Chapter 5 Class 12 Continuity and Differentiability
- Chapter 6 Class 12 Application of Derivatives
- Chapter 7 Class 12 Integrals
- Chapter 8 Class 12 Application of Integrals
- Chapter 9 Class 12 Differential Equations
- Chapter 10 Class 12 Vector Algebra
- Chapter 11 Class 12 Three Dimensional Geometry
- Chapter 12 Class 12 Linear Programming
- Chapter 13 Class 12 Probability

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.