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$Let g(x)=(2)/((x-3)^(2)). Select the correct description of the one-sided limits of g at x=3. Choose 1 answer: (A) {:[lim_(x rarr3^(+))g(x)=+oo" and "],[lim_(x rarr3^(-))g(x)=+oo]:} (B) {:[lim_(x rarr3^(+))g(x)=+oo" and "],[lim_(x rarr3^(-))g(x)=-oo]:} (c) {:[lim_(x rarr3^(+))g(x)=-oo" and "],[lim_(x rarr3^(-))g(x)=+oo]:} (D) {:[lim_(x rarr3^(+))g(x)=-oo" and "],[lim_(x rarr3^(-))g(x)=-oo]:}$

Let $g(x)=\frac{2}{(x-3)^{2}}$ . $\newline$ 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)^{2}}

\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}

Analyze function behavior: Analyze the function $g(x) = \frac{2}{(x - 3)^2}$ to understand its behavior near $x = 3$ . The function is a rational function with a denominator that approaches zero as $x$ approaches $3$ . Since the numerator is a positive constant ( $2$ ), the sign of $g(x)$ will depend on the sign of the denominator.
Limit from the right: Consider the limit as $x$ approaches $3$ from the right ( $x \to 3^+$ ). $\newline$ As $x$ gets closer to $3$ from the right, $(x - 3)^2$ is always positive because squaring a real number always gives a positive result. As the denominator approaches zero, the value of $g(x)$ becomes very large. Therefore, the limit from the right is positive infinity.
Limit from the left: Consider the limit as $x$ approaches $3$ from the left ( $x \to 3^-$ ). $\newline$ As $x$ gets closer to $3$ from the left, $(x - 3)^2$ is still always positive for the same reason as in Step $2$ . As the denominator approaches zero, the value of $g(x)$ becomes very large. Therefore, the limit from the left is also positive infinity.
Combine results: Combine the results from Step $2$ and Step $3$ to determine the one-sided limits of $g(x)$ at $x = 3$ . Since both one-sided limits are positive infinity, the correct description of the one-sided limits of $g$ at $x = 3$ is: $\lim_{x \to 3^+} g(x) = +\infty$ and $\lim_{x \to 3^-} g(x) = +\infty$ .