Let $f(x)=\frac{3}{x}$ . $\newline$ Select the correct description of the one-sided limits of $f$ at $x=0$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\lim _{x \rightarrow 0^{+}} f(x)=+\infty$ and $\lim _{x \rightarrow 0^{-}} f(x)=+\infty$ $\newline$ (B) $\lim _{x \rightarrow 0^{+}} f(x)=+\infty$ and $\lim _{x \rightarrow 0^{-}} f(x)=-\infty$ $\newline$ (C) $\lim _{x \rightarrow 0^{+}} f(x)=-\infty$ and $\lim _{x \rightarrow 0^{-}} f(x)=+\infty$ $\newline$ (D) $\lim _{x \rightarrow 0^{+}} f(x)=-\infty$ and $\lim _{x \rightarrow 0^{-}} f(x)=-\infty$

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Algebra 1

Compare linear and exponential growth

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Q. Let

f(x)=\frac{3}{x}

\newline

Select the correct description of the one-sided limits of

f

x=0

\newline

Choose

1

answer:

\newline

(A)

\lim _{x \rightarrow 0^{+}} f(x)=+\infty

and

\lim _{x \rightarrow 0^{-}} f(x)=+\infty

\newline

(B)

\lim _{x \rightarrow 0^{+}} f(x)=+\infty

and

\lim _{x \rightarrow 0^{-}} f(x)=-\infty

\newline

(C)

\lim _{x \rightarrow 0^{+}} f(x)=-\infty

and

\lim _{x \rightarrow 0^{-}} f(x)=+\infty

\newline

(D)

\lim _{x \rightarrow 0^{+}} f(x)=-\infty

and

\lim _{x \rightarrow 0^{-}} f(x)=-\infty

Analyze behavior of $f(x)$ as $x$ approaches $0+$ : Analyze the behavior of the function $f(x) = \frac{3}{x}$ as $x$ approaches $0$ from the positive side ( $x \rightarrow 0^+$ ). $\newline$ As $x$ gets closer to $0$ from the positive side, the value of $f(x)$ becomes larger and larger because the denominator of the fraction is getting smaller. This means that the function is approaching positive infinity.
Calculate limit of $f(x)$ as $x$ approaches $0+$ : Calculate the limit of $f(x)$ as $x$ approaches $0$ from the positive side. $\newline$ $\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \left(\frac{3}{x}\right) = +\infty$
Analyze behavior of $f(x)$ as $x$ approaches $0-$ : Analyze the behavior of the function $f(x) = \frac{3}{x}$ as $x$ approaches $0$ from the negative side ( $x \rightarrow 0^-$ ). $\newline$ As $x$ gets closer to $0$ from the negative side, the value of $f(x)$ becomes more negative and larger in magnitude because the denominator of the fraction is getting smaller and is negative. This means that the function is approaching negative infinity.
Calculate limit of $f(x)$ as $x$ approaches $0-$ : Calculate the limit of $f(x)$ as $x$ approaches $0$ from the negative side. $\newline$ $\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} \left(\frac{3}{x}\right) = -\infty$
Combine results from Step $2$ and Step $4$ : Combine the results from Step $2$ and Step $4$ to answer the question prompt. $\newline$ The one-sided limits of the function $f(x) = \frac{3}{x}$ as $x$ approaches $0$ are: $\newline$ $\lim_{x \to 0^+} f(x) = +\infty$ and $\lim_{x \to 0^-} f(x) = -\infty$