Determine whether the function f(x) is continuous at x=3. f(x)={[13-x^(2)",",x 3],[10-2x",",x

Question

Determine whether the function  f(x) is continuous at  x=3.  f(x)={[13-x^(2)",",x > 3],[10-2x",",x < 3]:}  f(x) is continuous at  x=3  f(x) is discontinuous at  x=3

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Check Function Definition: To determine if the function f(x) is continuous at x=3, we need to check three conditions: 1. The function is defined at x=3. 2. The limit of f(x) as x approaches 3 from the left (x -> 3-) is equal to the limit of f(x) as x approaches 3 from the right (x -> 3+). 3. The limit of f(x) as x approaches 3 is equal to the function value at x=3.Calculate Left Limit: First, let's check if the function is defined at x=3. We have two expressions for f(x), one for x > 3 and one for x < 3. To be defined at x=3, we need to have a value for f(3). Since the function is not explicitly defined at x=3, we cannot say it is defined at this point.Calculate Right Limit: Next, we calculate the limit of f(x) as x approaches 3 from the left (x -> 3-). This means we use the expression for f(x) when x < 3, which is f(x) = 10 - 2x. Limit as x -> 3- of f(x) = Limit as x -> 3- of (10 - 2x) = 10 - 2(3) = 10 - 6 = 4.Verify Function Value: Now, we calculate the limit of f(x) as x approaches 3 from the right (x -> 3+). This means we use the expression for f(x) when x > 3, which is f(x) = 13 - x^2. Limit as x -> 3+ of f(x) = Limit as x -> 3+ of (13 - x^2) = 13 - (3)^2 = 13 - 9 = 4.Function Continuity: Since the left-hand limit and the right-hand limit as x approaches 3 are both equal to 4, the second condition for continuity is satisfied. However, we still need to verify that the function value at x=3 is also 4 to satisfy the third condition.As we established in the second step, the function is not explicitly defined at x=3, which means we cannot find a function value f(3) to compare with the limits. Therefore, the function is not continuous at x=3 because it fails the third condition for continuity.

Determine whether the function $f(x)$ is continuous at $x=3$ . $\newline$ $f(x)=\left\{\begin{array}{ll} 13-x^{2}, & x>3 \\ 10-2 x, & x<3 \end{array}\right.$ $\newline$ $f(x)$ is continuous at $x=3$ $\newline$ $f(x)$ is discontinuous at $x=3$

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