Determine whether the function f(x) is continuous at x=6. f(x)={[16-x^(2)",",x = 6]:} f(x) is discontinuous at x=6 f(x) is continuous at x=6

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Determine whether the function  f(x) is continuous at  x=6.  f(x)={[16-x^(2)",",x < 6],[-2-3x",",x >= 6]:}  f(x) is discontinuous at  x=6  f(x) is continuous at  x=6

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Check Function Definition: To determine if the function f(x) is continuous at x=6, we need to check three conditions: 1. The function is defined at x=6. 2. The limit of f(x) as x approaches 6 exists. 3. The limit of f(x) as x approaches 6 is equal to the function value at x=6. Let's start by checking if the function is defined at x=6.Evaluate f(6): The function f(x) is defined piecewise, with one expression for x < 6 and another for x >= 6. At x=6, the function is defined by the second piece, f(x) = -2 - 3x. Let's evaluate f(6). f(6) = -2 - 3(6) = -2 - 18 = -20. So, the function is defined at x=6 and f(6) = -20.Calculate Left Limit: Next, we need to find the limit of f(x) as x approaches 6 from the left, which is denoted as lim(x->6^-) f(x). For x < 6, f(x) = 16 - x^2. Let's calculate the limit. lim(x->6^-) f(x) = lim(x->6^-) (16 - x^2) = 16 - (6)^2 = 16 - 36 = -20.Calculate Right Limit: Now, we need to find the limit of f(x) as x approaches 6 from the right, which is denoted as lim(x->6^+) f(x). For x >= 6, f(x) = -2 - 3x. Let's calculate the limit. lim(x->6^+) f(x) = lim(x->6^+) (-2 - 3x) = -2 - 3(6) = -2 - 18 = -20.Check Limit Existence: Since the limit from the left and the limit from the right both exist and are equal to each other, the limit of f(x) as x approaches 6 exists and is equal to -20.Compare Limit and Function Value: Finally, we compare the limit of f(x) as x approaches 6 to the function value at x=6. Since both the limit and the function value at x=6 are equal to -20, the function f(x) is continuous at x=6.

Determine whether the function $f(x)$ is continuous at $x=6$ . $\newline$ $f(x)=\left\{\begin{array}{ll} 16-x^{2}, & x<6 \\ -2-3 x, & x \geq 6 \end{array}\right.$ $\newline$ $f(x)$ is discontinuous at $x=6$ $\newline$ $f(x)$ is continuous at $x=6$

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