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

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

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Math Problems

Algebra 1

Compare linear, exponential, and quadratic growth

Full solution

Q. Let

g(x)=\frac{2}{x+3}

\newline

Select the correct description of the one-sided limits of

g

x=-3

\newline

Choose

1

answer:

\newline

(A)

\newline

\begin{array}{l} \lim _{x \rightarrow-3^{+}} g(x)=+\infty \text { and } \\ \lim _{x \rightarrow-3^{-}} g(x)=+\infty \end{array}

\newline

(B)

\newline

\begin{array}{l} \lim _{x \rightarrow-3^{+}} g(x)=+\infty \text { and } \\ \lim _{x \rightarrow-3^{-}} g(x)=-\infty \end{array}

\newline

(C)

\newline

\begin{array}{l} \lim _{x \rightarrow-3^{+}} g(x)=-\infty \text { and } \\ \lim _{x \rightarrow-3^{-}} g(x)=+\infty \end{array}

\newline

(D)

\newline

\begin{array}{l} \lim _{x \rightarrow-3^{+}} g(x)=-\infty \text { and } \\ \lim _{x \rightarrow-3^{-}} g(x)=-\infty \end{array}

Understand behavior near $-3$ : Analyze the function $g(x) = \frac{2}{(x+3)}$ to understand its behavior near $x = -3$ . As $x$ approaches $-3$ from the right ( $x > -3$ ), the denominator ( $x+3$ ) approaches $0$ from the positive side, making the fraction $\frac{2}{(x+3)}$ grow without bound in the positive direction.
Calculate right-hand limit: Calculate the right-hand limit of $g(x)$ as $x$ approaches $-3$ . $\lim_{x \to -3^{+}} g(x) = \lim_{x \to -3^{+}} \left(\frac{2}{x+3}\right) = +\infty$
Behavior near $-3$ from left: Analyze the function $g(x) = \frac{2}{x+3}$ to understand its behavior near $x = -3$ from the left side ( $x < -3$ ). $\newline$ As $x$ approaches $-3$ from the left, the denominator ( $x+3$ ) approaches $0$ from the negative side, making the fraction $\frac{2}{x+3}$ grow without bound in the negative direction.
Calculate left-hand limit: Calculate the left-hand limit of $g(x)$ as $x$ approaches $-3$ . $\lim_{x \to -3^{-}} g(x) = \lim_{x \to -3^{-}} \left(\frac{2}{x+3}\right) = -\infty$