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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 and exponential 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}

Behavior of Tangent Function: Understand the behavior of the tangent function near zero. $\newline$ The tangent function, $\tan(x)$ , approaches $0$ as $x$ approaches $0$ from both the positive and negative sides. Since $g(x) = \frac{1}{\tan^2(x)}$ , we need to consider what happens when we take the reciprocal of a number that is approaching zero.
Limit as $x$ Approaches $0$ from Positive Side: Consider the limit of $g(x)$ as $x$ approaches $0$ from the positive side. $\newline$ As $x$ approaches $0$ from the positive side, $\tan(x)$ approaches $0$ , and thus $\tan^2(x)$ also approaches $0$ . Taking the reciprocal of a positive number that is getting closer and closer to zero will result in a number that grows without bound. Therefore, the limit of $g(x)$ as $x$ approaches $0$ from the positive side is positive infinity. $\newline$ $0$ $4$
Limit as $x$ Approaches $0$ from Negative Side: Consider the limit of $g(x)$ as $x$ approaches $0$ from the negative side. $\newline$ As $x$ approaches $0$ from the negative side, $\tan(x)$ approaches $0$ , and thus $\tan^2(x)$ also approaches $0$ . Taking the reciprocal of a positive number that is getting closer and closer to zero will result in a number that grows without bound. Since $\tan^2(x)$ is positive whether $x$ approaches from the left or right, the limit from the negative side is also positive infinity. $\newline$ $0$ $3$