Determine whether the function f(x) is continuous at x=5. f(x)={[12-x^(2)",",x = 5]:} f(x) is continuous at x=5 f(x) is discontinuous at x=5

Question

Determine whether the function  f(x) is continuous at  x=5.  f(x)={[12-x^(2)",",x < 5],[-9-x",",x >= 5]:}  f(x) is continuous at  x=5  f(x) is discontinuous at  x=5

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Check Function Definition: To determine if the function f(x) is continuous at x=5, we need to check three conditions: 1. The function is defined at x=5. 2. The limit of f(x) as x approaches 5 exists. 3. The limit of f(x) as x approaches 5 is equal to the function value at x=5.Find Left-hand Limit: First, let's check if the function is defined at x=5. We have two expressions for f(x), one for x < 5 and one for x >= 5. Since 5 is included in the second expression, we will use that to find f(5). f(5) = -9 - 5 = -14. The function is defined at x=5.Find Right-hand Limit: Next, we need to find the limit of f(x) as x approaches 5 from the left, which we denote as lim(x->5^-) f(x). We use the expression for f(x) when x < 5, which is 12 - x^2. lim(x->5^-) f(x) = 12 - (5)^2 = 12 - 25 = -13.Determine Limit Existence: Now, we need to find the limit of f(x) as x approaches 5 from the right, which we denote as lim(x->5^+) f(x). We use the expression for f(x) when x >= 5, which is -9 - x. lim(x->5^+) f(x) = -9 - 5 = -14.Conclusion: We have found that the left-hand limit as x approaches 5 is -13 and the right-hand limit as x approaches 5 is -14. Since these two limits are not equal, the limit of f(x) as x approaches 5 does not exist. Therefore, the function f(x) is not continuous at x=5.

Determine whether the function $f(x)$ is continuous at $x=5$ . $\newline$ $f(x)=\left\{\begin{array}{ll} 12-x^{2}, & x<5 \\ -9-x, & x \geq 5 \end{array}\right.$ $\newline$ $f(x)$ is continuous at $x=5$ $\newline$ $f(x)$ is discontinuous at $x=5$

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