Determine whether the function f(x) is continuous at x=-2. f(x)={[9-5x^(2)",",x -2]:} f(x) is discontinuous at x=-2 f(x) is continuous at x=-2

Question

Determine whether the function  f(x) is continuous at  x=-2.  f(x)={[9-5x^(2)",",x <= -2],[-5+3x",",x > -2]:}  f(x) is discontinuous at  x=-2  f(x) is continuous at  x=-2

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Check Function Definition: To determine if the function f(x) is continuous at x=-2, we need to check three conditions: 1. The function is defined at x=-2. 2. The limit of f(x) as x approaches -2 from the left is equal to the limit of f(x) as x approaches -2 from the right. 3. The limit of f(x) as x approaches -2 is equal to the function value at x=-2.Find Left Limit: First, let's check if the function is defined at x=-2. We look at the piece of the function that applies when x <= -2, which is f(x) = 9 - 5x^2. Plugging in x=-2, we get f(-2) = 9 - 5(-2)^2 = 9 - 5(4) = 9 - 20 = -11. So, the function is defined at x=-2.Find Right Limit: Next, we need to find the limit of f(x) as x approaches -2 from the left. Since the function for x <= -2 is f(x) = 9 - 5x^2, the limit as x approaches -2 from the left is the same as the function value at x=-2, which we already calculated as -11.Verify Continuity: Now, we need to find the limit of f(x) as x approaches -2 from the right. For x > -2, the function is f(x) = -5 + 3x. Plugging in x=-2, we get the limit as x approaches -2 from the right to be -5 + 3(-2) = -5 - 6 = -11.Since the limit from the left and the limit from the right both equal -11, and the function value at x=-2 is also -11, all three conditions for continuity are satisfied. Therefore, the function f(x) is continuous at x=-2.

Determine whether the function $f(x)$ is continuous at $x=-2$ . $\newline$ $f(x)=\left\{\begin{array}{ll} 9-5 x^{2}, & x \leq-2 \\ -5+3 x, & x>-2 \end{array}\right.$ $\newline$ $f(x)$ is discontinuous at $x=-2$ $\newline$ $f(x)$ is continuous at $x=-2$

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