Determine whether the function f(x) is continuous at x=-2. f(x)={[12-3x^(2)",",x -2]:} f(x) is continuous at x=-2 f(x) is discontinuous at x=-2

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Determine whether the function  f(x) is continuous at  x=-2.  f(x)={[12-3x^(2)",",x <= -2],[-5-x",",x > -2]:}  f(x) is continuous at  x=-2  f(x) is discontinuous at  x=-2

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Check Conditions: To determine if the function f(x) is continuous at x=-2, we need to check if the following three conditions are met: 1. f(-2) is defined. 2. The limit of f(x) as x approaches -2 from the left side (lim x→-2⁻ f(x)) exists. 3. The limit of f(x) as x approaches -2 from the right side (lim x→-2⁺ f(x)) exists and is equal to the limit from the left side and to f(-2). First, we will find f(-2) using the piece of the function defined for x <= -2.Find f(-2): Substitute x = -2 into the first piece of the function, which is 12 - 3x^2, to find f(-2). f(-2) = 12 - 3(-2)^2 f(-2) = 12 - 3(4) f(-2) = 12 - 12 f(-2) = 0 So, f(-2) is defined and equals 0.Left Side Limit: Next, we need to find the limit of f(x) as x approaches -2 from the left side. Since the function for x <= -2 is 12 - 3x^2, we use this expression to find the limit. lim x→-2⁻ f(x) = lim x→-2⁻ (12 - 3x^2) lim x→-2⁻ f(x) = 12 - 3(-2)^2 lim x→-2⁻ f(x) = 12 - 12 lim x→-2⁻ f(x) = 0 The limit from the left side exists and equals 0.Right Side Limit: Now, we need to find the limit of f(x) as x approaches -2 from the right side. Since the function for x > -2 is -5 - x, we use this expression to find the limit. lim x→-2⁺ f(x) = lim x→-2⁺ (-5 - x) lim x→-2⁺ f(x) = -5 - (-2) lim x→-2⁺ f(x) = -5 + 2 lim x→-2⁺ f(x) = -3 The limit from the right side exists and equals -3.Function Continuity: Since the limit from the left side (0) is not equal to the limit from the right side (-3), the function f(x) is not continuous at x=-2. For a function to be continuous at a point, the limit from the left must equal the limit from the right and also equal the function value at that point. In this case, the limits do not match.

Determine whether the function $f(x)$ is continuous at $x=-2$ . $\newline$ $f(x)=\left\{\begin{array}{ll} 12-3 x^{2}, & x \leq-2 \\ -5-x, & x>-2 \end{array}\right.$ $\newline$ $f(x)$ is continuous at $x=-2$ $\newline$ $f(x)$ is discontinuous at $x=-2$

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