Determine whether the function f(x) is continuous at x=3. f(x)={[7-2x^(2)",",x 3]:} f(x) is continuous at x=3 Submit Answer f(x) is discontinuous at x=3

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Determine whether the function  f(x) is continuous at  x=3.  f(x)={[7-2x^(2)",",x <= 3],[-8-x",",x > 3]:}  f(x) is continuous at  x=3 Submit Answer  f(x) is discontinuous at  x=3

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Check Function Definition: To determine if the function f(x) is continuous at x=3, we need to check three conditions: 1. The function is defined at x=3. 2. The limit of f(x) as x approaches 3 from the left is equal to the limit of f(x) as x approaches 3 from the right. 3. The limit of f(x) as x approaches 3 is equal to the function value at x=3.Find Left Limit: First, let's check if the function is defined at x=3. We have two expressions for f(x), one for x <= 3 and one for x > 3. Since 3 is included in the domain of the first expression, we will use that to find f(3). f(3) = 7 - 2(3)^2 f(3) = 7 - 2(9) f(3) = 7 - 18 f(3) = -11 The function is defined at x=3 and f(3) = -11.Find Right Limit: Next, we need to find the limit of f(x) as x approaches 3 from the left. This means we will use the expression for f(x) when x <= 3. lim (x->3-) f(x) = lim (x->3-) (7 - 2x^2) lim (x->3-) f(x) = 7 - 2(3)^2 lim (x->3-) f(x) = 7 - 18 lim (x->3-) f(x) = -11Verify Continuity: Now, we need to find the limit of f(x) as x approaches 3 from the right. This means we will use the expression for f(x) when x > 3. lim (x->3+) f(x) = lim (x->3+) (-8 - x) lim (x->3+) f(x) = -8 - 3 lim (x->3+) f(x) = -11Since the limit from the left and the limit from the right both equal -11, and the function value at x=3 is also -11, all three conditions for continuity are satisfied. Therefore, f(x) is continuous at x=3.

Determine whether the function $f(x)$ is continuous at $x=3$ . $\newline$ $f(x)=\left\{\begin{array}{ll} 7-2 x^{2}, & x \leq 3 \\ -8-x, & x>3 \end{array}\right.$ $\newline$ $f(x)$ is continuous at $x=3$ $\newline$ Submit Answer $\newline$ $f(x)$ is discontinuous at $x=3$

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