Determine whether the function f(x) is continuous at x=-2. f(x)={[1-x^(2)",",x = -2],[-9-3x",",x

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Determine whether the function  f(x) is continuous at  x=-2.  f(x)={[1-x^(2)",",x >= -2],[-9-3x",",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. First, let's check if the function is defined at x=-2.Evaluate f(-2): The function f(x) is defined piecewise, with one expression for x >= -2 and another for x < -2. At x=-2, the function is defined by the first expression, which is f(x) = 1 - x^2. Let's evaluate f(-2) using the first expression: f(-2) = 1 - (-2)^2 f(-2) = 1 - 4 f(-2) = -3 The function is defined at x=-2 and f(-2) = -3.Find Left-hand Limit: Next, we need to find the limit of f(x) as x approaches -2 from the left, which means we use the expression for x < -2. The expression for x < -2 is f(x) = -9 - 3x. Let's find the limit as x approaches -2 from the left: lim (x -> -2-) f(x) = -9 - 3(-2) lim (x -> -2-) f(x) = -9 + 6 lim (x -> -2-) f(x) = -3 The left-hand limit as x approaches -2 is -3.Find Right-hand Limit: Now, we need to find the limit of f(x) as x approaches -2 from the right, which means we use the expression for x >= -2. The expression for x >= -2 is f(x) = 1 - x^2. Let's find the limit as x approaches -2 from the right: lim (x -> -2+) f(x) = 1 - (-2)^2 lim (x -> -2+) f(x) = 1 - 4 lim (x -> -2+) f(x) = -3 The right-hand limit as x approaches -2 is also -3.Verify Continuity: Since the left-hand limit and the right-hand limit as x approaches -2 are both equal to -3, and the function value at x=-2 is also -3, 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 $x=-2$ . $\newline$ $f(x)=\left\{\begin{array}{ll} 1-x^{2}, & x \geq-2 \\ -9-3 x, & x<-2 \end{array}\right.$ $\newline$ $f(x)$ is discontinuous at $x=-2$ $\newline$ $f(x)$ is continuous at $x=-2$

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