Example 7 - Is f(x) = |x| a continuous function - Class 12

Example 7 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Example 7 - Chapter 5 Class 12 Continuity and Differentiability - Part 3
Example 7 - Chapter 5 Class 12 Continuity and Differentiability - Part 4

  1. Chapter 5 Class 12 Continuity and Differentiability (Term 1)
  2. Concept wise

Transcript

Example 7 Is the function defined by f (x) = |x|, a continuous function? f(x) = |π‘₯| = {β–ˆ(βˆ’π‘₯, π‘₯<0@π‘₯, π‘₯β‰₯0)─ Since we need to find continuity at of the function We check continuity for different values of x When x = 0 When x < 0 When x > 0 Case 1 : When x = 0 f(x) is continuous at π‘₯ =0 if L.H.L = R.H.L = 𝑓(0) if lim┬(xβ†’0^βˆ’ ) 𝑓(π‘₯)=lim┬(xβ†’0^+ ) " " 𝑓(π‘₯)= 𝑓(0) Since there are two different functions on the left & right of 5, we take LHL & RHL . LHL at x β†’ 0 lim┬(xβ†’0^βˆ’ ) f(x) = lim┬(hβ†’0) f(0 βˆ’ h) = lim┬(hβ†’0) f(βˆ’h) = lim┬(hβ†’0) |βˆ’β„Ž| = lim┬(hβ†’0) β„Ž = 0 RHL at x β†’ 0 lim┬(xβ†’0^+ ) f(x) = lim┬(hβ†’0) f(0 + h) = lim┬(hβ†’0) f(h) = lim┬(hβ†’0) |β„Ž| = lim┬(hβ†’0) β„Ž = 0 & 𝑓(0) = 0 Hence, L.H.L = R.H.L = 𝑓(0) ∴ f is continuous at x=0 Case 2 : When x < 0 For x < 0, f(x) = βˆ’x Since this a polynomial It is continuous ∴ f(x) is continuous for x < 0 Case 3 : When x > 0 For x > 0, f(x) = x Since this a polynomial It is continuous ∴ f(x) is continuous for x > 0 Hence, 𝑓(π‘₯)= |π‘₯| is continuous at all points. i.e. f is continuous at 𝒙 ∈ R.

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.