Question

Determine whether the following function is continuous at a. Use the continuity checklist to justify your answer. (5x -2)/(x²-7x+10), a=2

Answer 1

The given function is not continuous at a=2 because the limit of the function as x approaches 2 does not exist.

Step-by-step explanation:

To determine whether the function (5x -2)/(x²-7x+10) is continuous at a=2, we can use the continuity checklist. The continuity checklist states that a function is continuous at a point if the following conditions are met:

1. The function is defined at a (i.e., there are no holes or vertical asymptotes at a).
2. The limit of the function as x approaches a exists.
3. The limit of the function as x approaches a is equal to the value of the function at a.

Let's check each condition for the given function:

1. The function is defined at a=2 because the denominator x²-7x+10 is not equal to zero when x=2.
2. The limit of the function as x approaches 2 can be found by evaluating the function at x=2. Substituting x=2 into the function, we get: (5(2)-2)/(2²-7(2)+10) = (10-2)/(4-14+10) = 8/0, which is undefined. Therefore, the limit as x approaches 2 does not exist.
3. Since the limit as x approaches 2 does not exist, the function is not continuous at a=2.

Therefore, the given function is not continuous at a=2.