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$Let g(x)=(1)/(tan^(2)(x)). Select the correct description of the one-sided limits of g at x=0. Choose 1 answer: (A) {:[lim_(x rarr0^(+))g(x)=+oo" and "],[lim_(x rarr0^(-))g(x)=+oo]:} (B) {:[lim_(x rarr0^(+))g(x)=+oo" and "],[lim_(x rarr0^(-))g(x)=-oo]:} (c) {:[lim_(x rarr0^(+))g(x)=-oo" and "],[lim_(x rarr0^(-))g(x)=+oo]:} (D) {:[lim_(x rarr0^(+))g(x)=-oo" and "],[lim_(x rarr0^(-))g(x)=-oo]:}$

Let $g(x)=\frac{1}{\tan ^{2}(x)}$ . $\newline$ Select the correct description of the one-sided limits of $g$ at $x=0$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\newline$ $\begin{array}{l} \lim _{x \rightarrow 0^{+}} g(x)=+\infty \text { and } \\ \lim _{x \rightarrow 0^{-}} g(x)=+\infty \end{array}$ $\newline$ (B) $\newline$ $\begin{array}{l} \lim _{x \rightarrow 0^{+}} g(x)=+\infty \text { and } \\ \lim _{x \rightarrow 0^{-}} g(x)=-\infty \end{array}$ $\newline$ (c) $\newline$ $\begin{array}{l} \lim _{x \rightarrow 0^{+}} g(x)=-\infty \text { and } \\ \lim _{x \rightarrow 0^{-}} g(x)=+\infty \end{array}$ $\newline$ (D) $\newline$ $\begin{array}{l} \lim _{x \rightarrow 0^{+}} g(x)=-\infty \text { and } \\ \lim _{x \rightarrow 0^{-}} g(x)=-\infty \end{array}$

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Math Problems

Algebra 1

Compare linear, exponential, and quadratic growth

Full solution

Q. Let

g(x)=\frac{1}{\tan ^{2}(x)}

\newline

Select the correct description of the one-sided limits of

g

x=0

\newline

Choose

1

answer:

\newline

(A)

\newline

\begin{array}{l} \lim _{x \rightarrow 0^{+}} g(x)=+\infty \text { and } \\ \lim _{x \rightarrow 0^{-}} g(x)=+\infty \end{array}

\newline

(B)

\newline

\begin{array}{l} \lim _{x \rightarrow 0^{+}} g(x)=+\infty \text { and } \\ \lim _{x \rightarrow 0^{-}} g(x)=-\infty \end{array}

\newline

(c)

\newline

\begin{array}{l} \lim _{x \rightarrow 0^{+}} g(x)=-\infty \text { and } \\ \lim _{x \rightarrow 0^{-}} g(x)=+\infty \end{array}

\newline

(D)

\newline

\begin{array}{l} \lim _{x \rightarrow 0^{+}} g(x)=-\infty \text { and } \\ \lim _{x \rightarrow 0^{-}} g(x)=-\infty \end{array}

Analyze Behavior of Function: We need to analyze the behavior of the function $g(x) = \frac{1}{\tan^2(x)}$ as $x$ approaches $0$ from both the right (positive) and left (negative) sides.
Limit as $x$ Approaches $0$ from Right: First, let's consider the limit as $x$ approaches $0$ from the right, which is denoted as $\lim_{x \to 0^+}g(x)$ . Since $\tan(x)$ approaches $0$ from the right as $x$ approaches $0$ , $\tan^2(x)$ also approaches $0$ . Therefore, $0$ $1$ approaches positive infinity because we are dividing by a positive number that is getting closer and closer to zero.
Limit as $x$ Approaches $0$ from Left: Now, let's consider the limit as $x$ approaches $0$ from the left, which is denoted as $\lim_{x \to 0^-}g(x)$ . Since $\tan(x)$ approaches $0$ from the left as $x$ approaches $0$ , $\tan^2(x)$ also approaches $0$ . Therefore, $0$ $1$ approaches positive infinity for the same reason as the right-hand limit: we are dividing by a positive number that is getting closer and closer to zero.
Conclusion: Since both one-sided limits as $x$ approaches $0$ are positive infinity, the correct description of the one-sided limits of $g$ at $x=0$ is that both are positive infinity.