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$Let g(x)=-(x)/(3sin(x-2)). Select the correct description of the one-sided limits of g at x=2. Choose 1 answer: (A) {:[lim_(x rarr2^(+))g(x)=+oo" and "],[lim_(x rarr2^(-))g(x)=+oo]:} (B) {:[lim_(x rarr2^(+))g(x)=+oo" and "],[lim_(x rarr2^(-))g(x)=-oo]:} (c) {:[lim_(x rarr2^(+))g(x)=-oo" and "],[lim_(x rarr2^(-))g(x)=+oo]:} (D) {:[lim_(x rarr2^(+))g(x)=-oo" and "],[lim_(x rarr2^(-))g(x)=-oo]:}$

Let $g(x)=-\frac{x}{3 \sin (x-2)}$ . $\newline$ Select the correct description of the one-sided limits of $g$ at $x=2$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\newline$ $\begin{array}{l} \lim _{x \rightarrow 2^{+}} g(x)=+\infty \text { and } \\ \lim _{x \rightarrow 2^{-}} g(x)=+\infty \end{array}$ $\newline$ (B) $\newline$ $\begin{array}{l} \lim _{x \rightarrow 2^{+}} g(x)=+\infty \text { and } \\ \lim _{x \rightarrow 2^{-}} g(x)=-\infty \end{array}$ $\newline$ (c) $\newline$ $\begin{array}{l} \lim _{x \rightarrow 2^{+}} g(x)=-\infty \text { and } \\ \lim _{x \rightarrow 2^{-}} g(x)=+\infty \end{array}$ $\newline$ (D) $\newline$ $\begin{array}{l} \lim _{x \rightarrow 2^{+}} g(x)=-\infty \text { and } \\ \lim _{x \rightarrow 2^{-}} g(x)=-\infty \end{array}$

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Math Problems

Algebra 1

Compare linear, exponential, and quadratic growth

Full solution

Q. Let

g(x)=-\frac{x}{3 \sin (x-2)}

\newline

Select the correct description of the one-sided limits of

g

x=2

\newline

Choose

1

answer:

\newline

(A)

\newline

\begin{array}{l} \lim _{x \rightarrow 2^{+}} g(x)=+\infty \text { and } \\ \lim _{x \rightarrow 2^{-}} g(x)=+\infty \end{array}

\newline

(B)

\newline

\begin{array}{l} \lim _{x \rightarrow 2^{+}} g(x)=+\infty \text { and } \\ \lim _{x \rightarrow 2^{-}} g(x)=-\infty \end{array}

\newline

(c)

\newline

\begin{array}{l} \lim _{x \rightarrow 2^{+}} g(x)=-\infty \text { and } \\ \lim _{x \rightarrow 2^{-}} g(x)=+\infty \end{array}

\newline

(D)

\newline

\begin{array}{l} \lim _{x \rightarrow 2^{+}} g(x)=-\infty \text { and } \\ \lim _{x \rightarrow 2^{-}} g(x)=-\infty \end{array}

Analyze Function Near $x=2$ : Analyze the function $g(x)$ near $x=2$ . We have the function $g(x) = -\frac{x}{3\sin(x-2)}$ . To find the one-sided limits as $x$ approaches $2$ , we need to consider the behavior of the sine function near $x=2$ .
Limit from the Right: Consider the limit from the right, $\lim_{x \to 2^+} g(x)$ . As $x$ approaches $2$ from the right, $(x-2)$ approaches $0$ from the right, and $\sin(x-2)$ approaches $\sin(0)$ from the right, which is $0$ . Since the sine function is continuous and smooth, just to the right of $0$ , the sine is positive. Therefore, $\sin(x-2)$ will be a small positive number, making $x$ $0$ a small positive number. The numerator $x$ $1$ will be a small negative number since $x$ is just slightly larger than $2$ . Dividing a small negative number by a small positive number yields a large negative number. Hence, $x$ $4$ .
Limit from the Left: Consider the limit from the left, $\lim_{x \to 2^-} g(x)$ . As $x$ approaches $2$ from the left, $(x-2)$ approaches $0$ from the left, and $\sin(x-2)$ approaches $\sin(0)$ from the left, which is $0$ . Since the sine function is continuous and smooth, just to the left of $0$ , the sine is negative. Therefore, $\sin(x-2)$ will be a small negative number, making $x$ $0$ a small negative number. The numerator $x$ $1$ will be a small negative number since $x$ is just slightly less than $2$ . Dividing a small negative number by another small negative number yields a large positive number. Hence, $x$ $4$ .