Check Full Chapter Explained - Continuity and Differentiability - https://you.tube/Chapter-5-Class-12-Continuity

Slide15.JPG

Slide16.JPG
Slide17.JPG Slide18.JPG

  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise

Transcript

Ex 5.1, 24 Determine if f defined by 𝑓(π‘₯)={β–ˆ( π‘₯2 sin⁑〖1/π‘₯γ€—, 𝑖𝑓 π‘₯β‰ 0@&0, 𝑖𝑓 π‘₯=0)─ is a continuous function? Since we need to find continuity at of the function We check continuity for different values of x When x β‰  0 When x = 0 Case 1 : When x β‰  0 For x β‰  0, f(x) = π‘₯2 sin⁑〖1/π‘₯γ€— Since x2 is continuous and sin⁑〖1/π‘₯γ€— is continuous So, π‘₯2 sin⁑〖1/π‘₯γ€— is continuous ∴ f(x) is continuous for x β‰  0 Case 2 : 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 0, 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)^2 sin⁑〖1/((βˆ’β„Ž))γ€— = lim┬(hβ†’0) β„Ž^2 π‘˜ = 02 .π‘˜ = 0 We know that βˆ’ 1 ≀ sin ΞΈ ≀ 1 β‡’ βˆ’ 1≀〖sin 〗⁑〖1/((βˆ’β„Ž))〗≀ 1 ∴ γ€–sin 〗⁑〖1/((βˆ’β„Ž))γ€— is a finite value Let γ€–sin 〗⁑〖1/((βˆ’β„Ž))γ€— = k 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^2 sin⁑〖1/β„Žγ€— = lim┬(hβ†’0) β„Ž^2 π‘˜ = 02 .π‘˜ = 0 We know that βˆ’ 1 ≀ sin ΞΈ ≀ 1 β‡’ βˆ’ 1≀〖sin 〗⁑〖1/β„Žγ€—β‰€ 1 ∴ γ€–sin 〗⁑〖1/β„Žγ€— is a finite value Let γ€–sin 〗⁑〖1/β„Žγ€— = k And, f(0) = 0 Hence, L.H.L = R.H.L = 𝑓(0) ∴ f is continuous at x=0 Hence, 𝒇(𝒙) is continuous for all real value of 𝒙 .

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.