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Example 4 - Show f(x) = {x3+3, 1 is not continuous at x = 0

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

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Example 4 Show that the function f given by 𝑓(π‘₯)={β–ˆ(π‘₯3+3, 𝑖𝑓 π‘₯β‰ [email protected], 𝑖𝑓 π‘₯=0)─ is not continuous at x = 0. f(x) is continuous at π‘₯ =0 if L.H.L = R.H.L = 𝑓(0) If (π‘™π‘–π‘š)┬(π‘₯β†’0^βˆ’ ) 𝑓(π‘₯)=(π‘™π‘–π‘š)┬(π‘₯β†’0^+ ) " " 𝑓(π‘₯)= 𝑓(0) Finding LHL and 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’h)3 + 3 = 03 + 3 = 3 RHL at x β†’ 0 lim┬(xβ†’0^+ ) f(x) = lim┬(hβ†’0) f(0 + h) = lim┬(hβ†’0) f(h) = lim┬(hβ†’0) h3 + 3 = 03 + 3 = 3 RHL at x β†’ 0 lim┬(xβ†’0^+ ) f(x) = lim┬(hβ†’0) f(0 + h) = lim┬(hβ†’0) f(h) = lim┬(hβ†’0) h3 + 3 = 03 + 3 = 3 But, f(0) = 1 So, LHL = RHL β‰  f(0) Hence, f is not continuous at 𝒙 = 𝟎

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.