Calculus AB

Which of the following functions is continuous at x = 3? (5 points)

Select one:
a. f of x equals the quotient of the quantity x squared minus 9 and the quantity x plus 3
b. f of x equals the quotient of the quantity x squared minus 9 and the quantity x minus 3 for x not equal to 3 and equals 3 for x equals 3
c. f of x equals the quotient of the quantity x squared minus 9 and the quantity x minus 3 for x not equal to 3 and equals 0 for x equals 3 Incorrect
d. All are continuous at x = 3

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  1. a) f(x) = (x^2 - 9)/(x+3)
    = (x^2 - 3^2)/(x+3)
    = (x+3)(x-3)/(x+3)
    = (x-3)

    Since this function is a linear one, it'll be continuous at all points.

    b) Using the similar identity as above,
    f(x) = (x+3), for x =/= 3
    Now, at all points except 3, the function is given by (x+3). So at points getting closer to 3, the function tends towards 6. But as per the question, f(3) = 3.
    Hence, it is discontinous at 3.

    c) This is the same as part (b), except f(3) equals 0 instead of 3. However, the same argument can be used to show its discontinuity.

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  2. (a) f is continuous at all points except x = -3, so it is continuous at x=3

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