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$f(x)={[(1)/(x+1)," for "-5 < x],[2^(x)," for "x > -4]:} Find lim_(x rarr-4^(+))f(x). Choose 1 answer: (A) -4 (B) -(1)/(3) (c) (1)/(16) (D) The limit doesn't exist.$

$f(x)=\left\{\begin{array}{ll}\frac{1}{x+1} & \text { for }-5<x \\="" 2^{x}="" &="" \text="" {="" for="" }="" x="">-4\end{array}\right.$ $\newline$ Find $\lim _{x \rightarrow-4^{+}} f(x)$ . $\newline$ Choose $1$ answer: $\newline$ (A) $-4$ $\newline$ (B) $-\frac{1}{3}$ $\newline$ (C) $\frac{1}{16}$ $\newline$ (D) The limit doesn't exist.

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Compare linear and exponential growth

Full solution

Q.

f(x)=\left\{\begin{array}{ll}\frac{1}{x+1} & \text { for }-5<x \\ 2^{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)

-\frac{1}{3}

\newline

(C)

\frac{1}{16}

\newline

(D) The limit doesn't exist.

Given function definition: We are given a piecewise function $f(x)$ and asked to find the limit as $x$ approaches $-4$ from the right ( $x \rightarrow -4^+$ ). The function is defined as follows: $\newline$ $f(x) = \begin{cases} \frac{1}{x+1} & \text{for } x > -5,\ 2^{x} & \text{for } x > -4 \end{cases}$ $\newline$ We need to determine which part of the piecewise function to use when calculating the limit as $x$ approaches $-4$ from the right.
Selecting appropriate part: Since we are looking for the limit as $x$ approaches $-4$ from the right, we will use the part of the function that is defined for x > -4, which is $f(x) = 2^{x}$ .
Calculating limit of selected part: Now we calculate the limit of $2^{x}$ as $x$ approaches $-4$ from the right: $\newline$ $\lim_{x \to -4^{+}} 2^{x}$ . $\newline$ Since $2^{x}$ is a continuous function for all $x$ , we can simply substitute $-4$ into the function to find the limit.
Substituting value and simplifying: Substituting $-4$ into the function, we get: $\newline$ $\lim_{x \to -4^{+}} 2^{x} = 2^{-4}.$
Final result: Calculating $2^{-4}$ , we get: $\newline$ $2^{-4} = 1/(2^4) = 1/16$ .
Final result: Calculating $2^{-4}$ , we get: $\newline$ $2^{-4} = 1/(2^4) = 1/16$ .Therefore, the limit of $f(x)$ as $x$ approaches $-4$ from the right is $1/16$ .

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\newline

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\newline

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