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 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  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.Find Left Limit: First, let's check if the function is defined at x=2. We look at the piece of the function that applies when x is greater than or equal to 2, which is f(x) = -8 + 3x. We substitute x with 2 to find f(2). f(2) = -8 + 3(2) = -8 + 6 = -2. The function is defined at x=2 because f(2) = -2.Find Right Limit: Next, we need to find the limit of f(x) as x approaches 2 from the left side (x < 2). We use the piece of the function that applies when x is less than 2, which is f(x) = 2 - x^2. We substitute x with a value that is very close to 2 from the left, such as 1.999. lim (x -> 2-) f(x) = 2 - (1.999)^2 = 2 - 3.996001 ≈ -1.996001. However, we need to consider the exact value as x approaches 2, not an approximation. lim (x -> 2-) f(x) = 2 - (2)^2 = 2 - 4 = -2.Verify Continuity: Now, we need to find the limit of f(x) as x approaches 2 from the right side (x >= 2). We use the piece of the function that applies when x is greater than or equal to 2, which is f(x) = -8 + 3x. We substitute x with a value that is very close to 2 from the right, such as 2.001. lim (x -> 2+) f(x) = -8 + 3(2.001) = -8 + 6.003 = -1.997. Again, we need to consider the exact value as x approaches 2, not an approximation. lim (x -> 2+) f(x) = -8 + 3(2) = -8 + 6 = -2.Since the limit from the left side as x approaches 2 is -2 and the limit from the right side as x approaches 2 is also -2, and both of these limits are equal to the function value at x=2 (f(2) = -2), 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 $x=2$ . $\newline$ $f(x)=\left\{\begin{array}{ll} 2-x^{2}, & x<2 \\ -8+3 x, & x \geq 2 \end{array}\right.$ $\newline$ $f(x)$ is continuous at $x=2$ $\newline$ $f(x)$ is discontinuous at $x=2$

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