Continuity of a Function

The continuity of a function at a specific point can be determined by checking three conditions:

1. The function must be defined at that point.
2. The limit of the function as x approaches that point must exist.
3. The value of the function at that point must be equal to the limit.

Determining the Continuity of f(x)

Let's analyze the function f(x) = (2|x|)/(3x - |x|) to determine its continuity at x = 1 and x = -1/2.

Continuity at x = 1

To check the continuity at x = 1, we need to evaluate the function at that point and check if it satisfies the three conditions mentioned earlier.

1. The function is defined at x = 1 since the denominator (3x - |x|) is nonzero.
2. We need to find the limit of the function as x approaches 1:
- As x approaches 1 from the left (x < 1),="" the="" expression="" simplifies="" to="" (2|x|)/(3x="" +="" x)="2/4" =="" />
- As x approaches 1 from the right (x > 1), the expression simplifies to (2|x|)/(3x - x) = 2/2 = 1.
- Therefore, the limit of the function as x approaches 1 exists.
3. We need to evaluate the function at x = 1:
- The expression simplifies to (2|1|)/(3 * 1 - |1|) = 2/2 = 1.
- The value of the function at x = 1 is equal to the limit.

Since all three conditions are satisfied, the function f(x) is continuous at x = 1.

Discontinuity at x = -1/2

To check the discontinuity at x = -1/2, we need to evaluate the function at that point and check if it satisfies the three conditions mentioned earlier.

1. The function is defined at x = -1/2 since the denominator (3x - |x|) is nonzero.
2. We need to find the limit of the function as x approaches -1/2:
- As x approaches -1/2 from the left (x < -1/2),="" the="" expression="" simplifies="" to="" (2|-x|)/(3x="" +="" x)="-2/2" =="" />
- As x approaches -1/2 from the right (x > -1/2), the expression simplifies to (2|x|)/(3x - x) = 2/2 = 1.
- Therefore, the limit of the function as x approaches -1/2 does not exist.
3. We need to evaluate the function at x = -1/2:
- The expression simplifies to (2|-(-1/2)|)/(3 * (-1/2) - |-1/2|) = 1/2 / (-3/2 + 1/2) = 1/2 / -2 = -1/4.
- The value of the function at x = -1/2 does not equal the limit.

Since the third condition is not satisfied, the function