If f x = e 2 x + 2 x - 1 2 x + 2 x : x ≠ 0 1 : x = 0 , then (where, [.] represents the greatest integer function)

Question

If f x = e 2 x + 2 x - 1 2 x + 2 x : x ≠ 0 1 : x = 0 , then (where, [.] represents the greatest integer function), l i m x → 0 + f x = - 1, l i m x → 0 - f x = 1 e - 1, f x is differentiable at x = 0, f x is discontinuous at x = 0

MARKS App · Accepted Answer

l i m x → 0 + f x = l i m x → 0 + f 0 + h = l i m h → 0 + e 2 h + 2 h - 1 2 h + 2 h = l i m h → 0 + e 2 h - 1 2 h = 1 and l i m x → 0 - f x = l i m h → 0 + f 0 - h = l i m h → 0 e - 2 h - 2 h - 1 - 2 h - 2 h = l i m h → 0 e - 1 - 2 h - 1 - 1 - 2 h = e - 1 - 1 - 1 = 1 - 1 e Since, l i m x → 0 + f x ≠ l i m x → 0 - f x , therefore, f x is not continuous at x = 0 .

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