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

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Example 4 Show that the function f given by 𝑓﷐𝑥﷯=﷐﷐𝑥3+3, 𝑖𝑓 𝑥≠0﷮1, 𝑖𝑓 𝑥=0﷯﷯ is not continuous at x = 0. Given, 𝑓﷐𝑥﷯=﷐﷐𝑥3+3, 𝑖𝑓 𝑥≠0﷮1, 𝑖𝑓 𝑥=0﷯﷯ We need to show that f is not continuous at 𝑥=0 f is continuous at 𝑥 = 0 if, L.H.L = R.H.L = 𝑓(0) i.e. ﷐lim﷮x→﷐0﷮−﷯﷯ 𝑓﷐𝑥﷯=﷐lim﷮x→﷐0﷮+﷯﷯ 𝑓﷐𝑥﷯=𝑓(0) But, 𝑓(0) =1 ≠3 So, ﷐lim﷮x→﷐0﷮+﷯﷯ 𝑓﷐𝑥﷯=﷐lim﷮x→﷐0﷮−﷯﷯ 𝑓﷐𝑥﷯≠𝑓(0) Hence, f is not continuous at 𝒙 = 𝟎

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