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$f(x)={[2^(x)," for "0 < x <= 4],[8sqrtx," for "x > 4]:} Find lim_(x rarr4)f(x). Choose 1 answer: (A) 4 (B) 8 (c) 16 (D) The limit doesn't exist.$

$f(x)=\left\{\begin{array}{ll} 2^{x} & \text { for } 0<x \leq 4 \\ 8 \sqrt{x} & \text { for } x>4 \end{array}\right.$ $\newline$ Find $\lim _{x \rightarrow 4} f(x)$ . $\newline$ Choose $1$ answer: $\newline$ (A) $4$ $\newline$ (B) $8$ $\newline$ (C) $16$ $\newline$ (D) The limit doesn't exist.

Full solution

f(x)=\left\{\begin{array}{ll} 2^{x} & \text { for } 0<x \leq 4 \\ 8 \sqrt{x} & \text { for } x>4 \end{array}\right.

\newline

Find

\lim _{x \rightarrow 4} f(x)

\newline

Choose

1

answer:

\newline

(A)

4

\newline

(B)

8

\newline

(C)

16

\newline

(D) The limit doesn't exist.

Given Function: We are given a piecewise function $f(x)$ and need to find the limit as $x$ approaches $4$ . The function is defined differently for two intervals: when $0 < x \leq 4$ , $f(x) = 2^x$ , and when $x > 4$ , $f(x) = 8\sqrt{x}$ . To find the limit as $x$ approaches $4$ , we need to consider the value of the function as $x$ approaches $4$ from both the left and the right.
Limit from the Left: First, let's find the limit from the left, which means we are considering the interval where $0 < x \leq 4$ . In this interval, the function is defined as $f(x) = 2^x$ . So, we need to calculate the limit of $2^x$ as $x$ approaches $4$ from the left. $\newline$ $\lim_{x \to 4-} 2^x = 2^4 = 16$
Limit from the Right: Now, let's find the limit from the right, which means we are considering the interval where $x > 4$ . In this interval, the function is defined as $f(x) = 8\sqrt{x}$ . However, since we are only interested in the limit as $x$ approaches $4$ , we need to consider the value of the function as $x$ gets infinitely close to $4$ from the right. $\newline$ $\lim_{x \to 4^+} 8\sqrt{x} = 8\sqrt{4} = 8 \times 2 = 16$
Existence of Limit: Since the limit from the left and the limit from the right are equal, the limit of the function as $x$ approaches $4$ exists and is equal to the common value. $\lim_{x \to 4} f(x) = 16$