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Example 3 - Discuss continuity of f(x) = |x| at x = 0 - Class 12

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

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Example 3 Discuss the continuity of the function f given by 𝑓(π‘₯) =|π‘₯| π‘Žπ‘‘ π‘₯ = 0. 𝑓(π‘₯) = |π‘₯| 𝑓(π‘₯)= {β–ˆ(βˆ’π‘₯, 𝑖𝑓 π‘₯<0@π‘₯, 𝑖𝑓 π‘₯ β‰₯0)─ f is continuous at π‘₯ = 0 if L.H.L = R.H.L = 𝑓(0) i.e. (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| = lim┬(hβ†’0) h = 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) \h| = lim┬(hβ†’0) h = 0 And, f(0) = 0 So, LHL = RHL = f(0) Hence, f is not continuous at 𝒙 = 𝟎

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