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

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Determine whether the function  f(x) is continuous at  x=3.  f(x)={[10-3x^(2)",",x < -3],[-5+3x",",x >= -3]:}  f(x) is discontinuous at  x=3  f(x) is continuous 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 if the following three conditions are met: 1. The function is defined at x=3. 2. The limit of f(x) as x approaches 3 exists. 3. The limit of f(x) as x approaches 3 is equal to the function value at x=3.Find Left Limit: First, let's check if the function is defined at x=3. We look at the piece of the function that applies when x is greater than or equal to -3, which is f(x) = -5 + 3x. Plugging in x=3, we get f(3) = -5 + 3(3) = -5 + 9 = 4. So, the function is defined at x=3.Find Right Limit: Next, we need to find the limit of f(x) as x approaches 3 from the left and from the right. For x approaching 3 from the left, we use the piece of the function for x >= -3, which is f(x) = -5 + 3x. The limit as x approaches 3 from the left is the same as the function value at x=3, which we already calculated as 4.Verify Continuity: Now, we find the limit of f(x) as x approaches 3 from the right. Since there is no piece of the function defined for x > 3, we use the same piece as for the left-hand limit, f(x) = -5 + 3x. The limit as x approaches 3 from the right is also 4.Since the left-hand limit and the right-hand limit as x approaches 3 are both equal to 4, and the function value at x=3 is also 4, all three conditions for continuity are satisfied. Therefore, the function f(x) is continuous at x=3.

Determine whether the function $f(x)$ is continuous at $x=3$ . $\newline$ $f(x)=\left\{\begin{array}{ll} 10-3 x^{2}, & x<-3 \\ -5+3 x, & x \geq-3 \end{array}\right.$ $\newline$ $f(x)$ is discontinuous at $x=3$ $\newline$ $f(x)$ is continuous at $x=3$

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