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

$f(x)=\left\{\begin{array}{ll}\frac{1}{x+1} & \text { for }-5<x \leq-4="" \\="" 2^{x}="" &="" \text="" {="" for="" }="" x="">-4\end{array}\right.$ $\newline$ 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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Full solution

Q.

f(x)=\left\{\begin{array}{ll}\frac{1}{x+1} & \text { for }-5<x \leq-4 \\ 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.

Define Function: We need to find the limit of $f(x)$ as $x$ approaches $-4$ from the right, which is denoted as $\lim_{x \to -4^{+}}f(x)$ . To do this, we look at the definition of the function $f(x)$ for values of $x$ that are greater than $-4$ .
Use Defined Function: Since we are looking for the limit as $x$ approaches $-4$ from the right, we use the piece of the function that is defined for x > -4, which is $f(x) = 2^{x}$ .
Substitute $x$ Value: We substitute $x$ with $-4$ in the expression $2^{x}$ to find the limit as $x$ approaches $-4$ from the right. $\newline$ $\lim_{x \rightarrow -4^{+}}f(x) = \lim_{x \rightarrow -4^{+}}2^{x} = 2^{-4}$
Calculate Limit: Now we calculate $2^{-4}$ , which is the same as $1/(2^4)$ . $2^{-4} = 1/(2^4) = 1/16$
Final Answer: The limit of $f(x)$ as $x$ approaches $-4$ from the right is $\frac{1}{16}$ . Therefore, the correct answer is (C) $\frac{1}{16}$ .

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