Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

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

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

View step-by-step help

Home

Math Problems

Algebra 1

Compare linear and exponential 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}

Define function: $g(x) = \frac{2}{(x-3)^{2}}$ . Let's find the limit as $x$ approaches $3$ from the right ( $x \to 3^+$ ).
Limit from right: As $x$ gets closer to $3$ from the right, $(x-3)$ gets closer to $0$ , making $(x-3)^2$ also get closer to $0$ . Since we're dividing $2$ by a number that's getting closer to $0$ , the function value will increase without bound.
Limit from left: $\lim_{x \to 3^+} g(x) = +\infty$ .
Conclusion: Now let's find the limit as $x$ approaches $3$ from the left ( $x \to 3^-$ ).
Conclusion: Now let's find the limit as $x$ approaches $3$ from the left ( $x \to 3^-$ ).As $x$ gets closer to $3$ from the left, $(x-3)$ gets closer to $0$ , making $(x-3)^2$ also get closer to $0$ . Since we're dividing $2$ by a number that's getting closer to $0$ , the function value will increase without bound, just like from the right side.
Conclusion: Now let's find the limit as $x$ approaches $3$ from the left ( $x \to 3^-$ ).As $x$ gets closer to $3$ from the left, $(x-3)$ gets closer to $0$ , making $(x-3)^2$ also get closer to $0$ . Since we're dividing $2$ by a number that's getting closer to $0$ , the function value will increase without bound, just like from the right side. $3$ $1$
Conclusion: Now let's find the limit as $x$ approaches $3$ from the left ( $x \to 3^-$ ).As $x$ gets closer to $3$ from the left, $(x-3)$ gets closer to $0$ , making $(x-3)^2$ also get closer to $0$ . Since we're dividing $2$ by a number that's getting closer to $0$ , the function value will increase without bound, just like from the right side. $3$ $1$ .Both one-sided limits as $x$ approaches $3$ are $3$ $4$ , so the correct answer is $3$ $5$ $\left\{:\left[\lim_{x \to\(3$^{+}}g(x)=+\infty\quad \text{and} \quad\right],\left[\lim_{x \to$3$^{-}}g(x)=+\infty\right]:\right\}\.}