Determine whether the function f(x) is continuous at x=5. f(x)={[12-x^(2)",",x = 5]:} f(x) is continuous at x=5 Submit Answer f(x) is discontinuous at x=5

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Determine whether the function  f(x) is continuous at  x=5.  f(x)={[12-x^(2)",",x < 5],[-9-x",",x >= 5]:}  f(x) is continuous at  x=5 Submit Answer  f(x) is discontinuous at  x=5

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Check Conditions: To determine if the function f(x) is continuous at x=5, we need to check if the following three conditions are met: 1. The function is defined at x=5. 2. The limit of f(x) as x approaches 5 from the left (lim x→5⁻ f(x)) equals the limit of f(x) as x approaches 5 from the right (lim x→5⁺ f(x)). 3. The limit of f(x) as x approaches 5 equals the function value at x=5 (f(5)).Find f(5): First, let's find the function value at x=5. Since 5 is not less than 5, we use the piece of the function defined for x >= 5, which is f(x) = -9 - x. f(5) = -9 - 5 = -14.Calculate lim x→5⁻ f(x): Next, we calculate the limit of f(x) as x approaches 5 from the left. For x < 5, the function is defined as f(x) = 12 - x^2. lim x→5⁻ f(x) = lim x→5⁻ (12 - x^2) = 12 - (5)^2 = 12 - 25 = -13.Calculate lim x→5⁺ f(x): Now, we calculate the limit of f(x) as x approaches 5 from the right. For x >= 5, the function is defined as f(x) = -9 - x. lim x→5⁺ f(x) = lim x→5⁺ (-9 - x) = -9 - 5 = -14.Compare Limits: We compare the left-hand limit, right-hand limit, and the function value at x=5. We found that: lim x→5⁻ f(x) = -13 lim x→5⁺ f(x) = -14 f(5) = -14 Since the left-hand limit does not equal the right-hand limit, the function is not continuous at x=5.

Determine whether the function $f(x)$ is continuous at $x=5$ . $\newline$ $f(x)=\left\{\begin{array}{ll} 12-x^{2}, & x<5 \\ -9-x, & x \geq 5 \end{array}\right.$ $\newline$ $f(x)$ is continuous at $x=5$ $\newline$ Submit Answer $\newline$ $f(x)$ is discontinuous at $x=5$

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