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

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Determine whether the function  f(x) is continuous at  x=2.  f(x)={[2-x^(2)",",x < 2],[-8+3x",",x >= 2]:}  f(x) is continuous at  x=2 Submit Answer  f(x) is discontinuous at  x=2

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Definition of Continuity: Understand the definition of continuity at a point. A function f(x) is continuous at a point x=a if the following three conditions are met: 1. f(a) is defined. 2. The limit of f(x) as x approaches a exists. 3. The limit of f(x) as x approaches a is equal to f(a).Check Function Definition: Check if f(2) is defined for the given function. The function f(x) is defined piecewise, so we need to check both pieces to see which one applies at x=2. Since x=2 falls in the second piece of the function, we use the expression for x >= 2 to find f(2). f(2) = -8 + 3(2) = -8 + 6 = -2 So, f(2) is defined and equals -2.Left-hand Limit: Find the limit of f(x) as x approaches 2 from the left. For x < 2, the function is defined as f(x) = 2 - x^2. We need to find the limit as x approaches 2 from the left. lim (x -> 2-) f(x) = lim (x -> 2-) (2 - x^2) = 2 - (2)^2 = 2 - 4 = -2Right-hand Limit: Find the limit of f(x) as x approaches 2 from the right. For x >= 2, the function is defined as f(x) = -8 + 3x. We need to find the limit as x approaches 2 from the right. lim (x -> 2+) f(x) = lim (x -> 2+) (-8 + 3x) = -8 + 3(2) = -8 + 6 = -2Comparison of Limits: Compare the left-hand limit, right-hand limit, and the value of f(2). We have found that: lim (x -> 2-) f(x) = -2 lim (x -> 2+) f(x) = -2 f(2) = -2 Since all three values are equal, the function f(x) is continuous at x=2.

Determine whether the function $f(x)$ is continuous at $x=2$ . $\newline$ $f(x)=\left\{\begin{array}{ll} 2-x^{2}, & x<2 \\ -8+3 x, & x \geq 2 \end{array}\right.$ $\newline$ $f(x)$ is continuous at $x=2$ $\newline$ Submit Answer $\newline$ $f(x)$ is discontinuous at $x=2$

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