Determine whether the function f(x) is continuous at x=2. f(x)={[6+3x^(2)",",x = 2],[20-2x",",x

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Determine whether the function  f(x) is continuous at  x=2.  f(x)={[6+3x^(2)",",x >= 2],[20-2x",",x < 2]:}  f(x) is continuous at  x=2  f(x) is discontinuous 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 exists. 3. The limit of f(x) as x approaches 2 is equal to the function value at x=2. Let's start by checking if the function is defined at x=2.Evaluate f(2): We have two expressions for f(x), one for x >= 2 and one for x < 2. Since we are checking continuity at x=2, we need to evaluate the function at x=2 using the expression for x >= 2. f(2) = 6 + 3(2)^2 f(2) = 6 + 3(4) f(2) = 6 + 12 f(2) = 18 The function is defined at x=2 and f(2) = 18.Find Left Limit: Next, we need to find the limit of f(x) as x approaches 2 from the left side (x < 2). We use the expression for f(x) when x < 2. lim (x -> 2-) f(x) = lim (x -> 2-) (20 - 2x) lim (x -> 2-) f(x) = 20 - 2(2) lim (x -> 2-) f(x) = 20 - 4 lim (x -> 2-) f(x) = 16 The limit of f(x) as x approaches 2 from the left is 16.Find Right Limit: Now, we need to find the limit of f(x) as x approaches 2 from the right side (x >= 2). We use the expression for f(x) when x >= 2. lim (x -> 2+) f(x) = lim (x -> 2+) (6 + 3x^2) lim (x -> 2+) f(x) = 6 + 3(2)^2 lim (x -> 2+) f(x) = 6 + 3(4) lim (x -> 2+) f(x) = 6 + 12 lim (x -> 2+) f(x) = 18 The limit of f(x) as x approaches 2 from the right is 18.Determine Continuity: We have found that the limit from the left is 16 and the limit from the right is 18. Since these two limits are not equal, the limit of f(x) as x approaches 2 does not exist. Therefore, the function is not continuous at x=2.

Determine whether the function $f(x)$ is continuous at $x=2$ . $\newline$ $f(x)=\left\{\begin{array}{ll} 6+3 x^{2}, & x \geq 2 \\ 20-2 x, & x<2 \end{array}\right.$ $\newline$ $f(x)$ is continuous at $x=2$ $\newline$ $f(x)$ is discontinuous at $x=2$

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