Which of the following function is discontinuous at x = 0 ?

Step-by-step solution

Let us check the continuity of each function at x = 0 by evaluating the limits. For the first function: _ x 0 (1 + x)^ 2 x = e^ _ x 0 2 x (1+x) = e^2 = f(0) The function is continuous at x = 0 . For the second function: _ x 0 ( x - x) = 0 - 0 = -1 = f(0) The function is continuous at x = 0 . For the third function: Right-hand limit (RHL): _ x 0^+ e^ 1 x - 1 e^ 1 x + 1 = _ x 0^+ 1 - e^ - 1 x 1 + e^ - 1 x = 1 - 0 1 + 0 = 1 Left-hand limit (LHL): _ x 0^- e^ 1 x - 1 e^ 1 x + 1 = 0 - 1 0 + 1 = -1 Since LHL RHL, the limi