Final answer:
The function f(x) = 6x/(x + 2) is continuous at x = 2 but not at x = -2 because the function is undefined at x = -2 due to division by zero.
Step-by-step explanation:
The function f(x) = 6x/(x + 2) needs to be analyzed for continuity at x = 2 and x = -2. A function is said to be continuous at a point if the following three conditions are met: the function is defined at the point, the limit as x approaches the point exists, and the limit equals the function's value at that point.
For x = 2, substituting the value into the function gives f(2) = 6(2)/(2 + 2) = 12/4 = 3, and limits as x approaches 2 from both left and right also result in the same value. Therefore, the function f(x) is continuous at x = 2.
However, for x = -2, there is a problem. Substituting x = -2 into the denominator of the function gives 0, which indicates a division by zero, making the function undefined at this point. Hence, f(x) is not continuous at x = -2 because the function is not defined there.
Thus, f(x) is continuous at x = 2 but not at x = -2.