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$Determine whether the function f(x) is continuous at x=3. f(x)={[7-x^(2)",",x < 3],[-6+x",",x >= 3]:} f(x) is discontinuous at x=3 f(x) is continuous at x=3$

Determine whether the function $f(x)$ is continuous at $x=3$ . $\newline$ $f(x)=\left\{\begin{array}{ll} 7-x^{2}, & x<3 \\ -6+x, & x \geq 3 \end{array}\right.$ $\newline$ $f(x)$ is discontinuous at $x=3$ $\newline$ $f(x)$ is continuous at $x=3$

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Identify discrete and continuous random variables

Full solution

Q. Determine whether the function

f(x)

is continuous at

x=3

.

\newline

f(x)=\left\{\begin{array}{ll} 7-x^{2}, & x<3 \\ -6+x, & x \geq 3 \end{array}\right.

\newline

f(x)

is discontinuous at

x=3

\newline

f(x)

is continuous at

x=3

Check Function Definition: To determine if the function $f(x)$ is continuous at $x=3$ , we need to check three conditions: $\newline$ $1$ . The function must be defined at $x=3$ . $\newline$ $2$ . The limit of $f(x)$ as $x$ approaches $3$ must exist. $\newline$ $3$ . The limit of $f(x)$ as $x$ approaches $3$ must be equal to the function value at $x=3$ . $\newline$ First, let's check if the function is defined at $x=3$ . $\newline$ We look at the piece of the function that applies when $x$ is greater than or equal to $3$ , which is $x=3$ $3$ . $\newline$ We evaluate the function at $x=3$ : $x=3$ $5$ . $\newline$ The function is defined at $x=3$ .
Find Left Limit: Next, we need to find the limit of $f(x)$ as $x$ approaches $3$ from the left side ( $x < 3$ ). $\newline$ For $x < 3$ , the function is defined as $f(x) = 7 - x^2$ . $\newline$ We calculate the limit as $x$ approaches $3$ from the left: $\lim_{x\to3^-} f(x) = 7 - (3)^2 = 7 - 9 = -2$ .
Find Right Limit: Now, we need to find the limit of $f(x)$ as $x$ approaches $3$ from the right side ( $x \geq 3$ ). $\newline$ For $x \geq 3$ , the function is defined as $f(x) = -6 + x$ . $\newline$ We calculate the limit as $x$ approaches $3$ from the right: $\lim_{x\to3^+} f(x) = -6 + 3 = -3$ .
Compare Limits: Finally, we compare the two one-sided limits and the function value at $x=3$ . The left-hand limit as $x$ approaches $3$ is $-2$ , and the right-hand limit as $x$ approaches $3$ is $-3$ . Since the left-hand limit does not equal the right-hand limit, the limit of $f(x)$ as $x$ approaches $3$ does not exist. Therefore, the function $f(x)$ is not continuous at $x=3$ .

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