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$Determine whether the function f(x) is continuous at x=4. f(x)={[11-2x^(2)",",x <= 4],[-16-x",",x > 4]:} f(x) is discontinuous at x=4 f(x) is continuous at x=4$

Determine whether the function $f(x)$ is continuous at $x=4$ . $\newline$ $f(x)=\left\{\begin{array}{ll} 11-2 x^{2}, & x \leq 4 \\ -16-x, & x>4 \end{array}\right.$ $\newline$ $f(x)$ is discontinuous at $x=4$ $\newline$ $f(x)$ is continuous at $x=4$

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Math Problems

Calculus

Intermediate Value Theorem

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Q. Determine whether the function

f(x)

is continuous at

x=4

\newline

f(x)=\left\{\begin{array}{ll} 11-2 x^{2}, & x \leq 4 \\ -16-x, & x>4 \end{array}\right.

\newline

f(x)

is discontinuous at

x=4

\newline

f(x)

is continuous at

x=4

Check Conditions: To determine if the function $f(x)$ is continuous at $x=4$ , we need to check if the following three conditions are met: $\newline$ $1$ . $f(4)$ is defined. $\newline$ $2$ . The limit of $f(x)$ as $x$ approaches $4$ from the left side ( $\lim_{x\to 4^-} f(x)$ ) equals the limit of $f(x)$ as $x$ approaches $4$ from the right side ( $x=4$ $0$ ). $\newline$ $3$ . Both of these limits equal $f(4)$ .
Find $f(4)$ : First, we find $f(4)$ using the piece of the function defined for $x \leq 4$ , which is $f(x) = 11 - 2x^2$ . Substitute $x = 4$ into this piece to get $f(4) = 11 - 2(4)^2 = 11 - 32 = -21$ .
Find $\lim_{x \to 4^-} f(x)$ : Next, we find the limit of $f(x)$ as $x$ approaches $4$ from the left side ( $\lim_{x \to 4^-} f(x)$ ). $\newline$ Since the function for $x \leq 4$ is $f(x) = 11 - 2x^2$ , the limit as $x$ approaches $4$ from the left is the same as $f(4)$ , which is $f(x)$ $0$ .
Find $\lim_{x \to 4^+} f(x)$ : Now, we find the limit of $f(x)$ as $x$ approaches $4$ from the right side ( $\lim_{x \to 4^+} f(x)$ ). $\newline$ For $x > 4$ , the function is defined as $f(x) = -16 - x$ . We substitute $x = 4$ into this piece to find the limit as $x$ approaches $4$ from the right, which gives us $f(x)$ $0$ .
Compare Limits: We compare the two one-sided limits and the value of $f(4)$ . From the left, the limit is $-21$ , and from the right, the limit is $-20$ . Since these two limits are not equal, the function is not continuous at $x=4$ .