Find at least one point at which each of the following functions is not continuous and state which of the three conditions in the definition of continuity is violated at that point:
a) (x+5)/(x+3)
b) (x^2 + x + 6) / (x-2)
c) sqrt(cosx)
d) |x|^2
e) x/sin(x)
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a) The function (x+5)/(x+3) is not continuous at x = -3 because the denominator becomes zero at that point. This violates the third condition of the definition of continuity, which states that a function must not have a point of discontinuity.
b) The function (x^2 + x + 6) / (x-2) is not continuous at x = 2 because the denominator becomes zero at that point. This violates the third condition of the definition of continuity.
c) The function sqrt(cosx) is not continuous at x = (2n + 1)π/2, where n is an integer. At these points, the function is not defined because the cosine function is zero, resulting in the square root of zero. This violates the first condition of the definition of continuity, which states that a function must be defined at a given point to be continuous at that point.
d) The function |x|^2 is not continuous at x = 0 because the function changes abruptly in value at that point. Specifically, the function is equal to 0 for x < 0 and equal to x^2 for x > 0. This violates the second condition of the definition of continuity, which states that the limit of the function must exist at a given point to be continuous at that point.
e) The function x/sin(x) is not continuous at x = 0 because the denominator, sin(x), becomes zero at that point. This violates the third condition of the definition of continuity.
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