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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)=1tan2(x) g(x)=\frac{1}{\tan ^{2}(x)} .\newlineSelect the correct description of the one-sided limits of g g at x=0 x=0 .\newlineChoose 11 answer:\newline(A)\newlinelimx0+g(x)=+ and limx0g(x)=+ \begin{array}{l} \lim _{x \rightarrow 0^{+}} g(x)=+\infty \text { and } \\ \lim _{x \rightarrow 0^{-}} g(x)=+\infty \end{array} \newline(B)\newlinelimx0+g(x)=+ and limx0g(x)= \begin{array}{l} \lim _{x \rightarrow 0^{+}} g(x)=+\infty \text { and } \\ \lim _{x \rightarrow 0^{-}} g(x)=-\infty \end{array} \newline(C)\newlinelimx0+g(x)= and limx0g(x)=+ \begin{array}{l} \lim _{x \rightarrow 0^{+}} g(x)=-\infty \text { and } \\ \lim _{x \rightarrow 0^{-}} g(x)=+\infty \end{array} \newline(D)\newlinelimx0+g(x)= and limx0g(x)= \begin{array}{l} \lim _{x \rightarrow 0^{+}} g(x)=-\infty \text { and } \\ \lim _{x \rightarrow 0^{-}} g(x)=-\infty \end{array}

Full solution

Q. Let g(x)=1tan2(x) g(x)=\frac{1}{\tan ^{2}(x)} .\newlineSelect the correct description of the one-sided limits of g g at x=0 x=0 .\newlineChoose 11 answer:\newline(A)\newlinelimx0+g(x)=+ and limx0g(x)=+ \begin{array}{l} \lim _{x \rightarrow 0^{+}} g(x)=+\infty \text { and } \\ \lim _{x \rightarrow 0^{-}} g(x)=+\infty \end{array} \newline(B)\newlinelimx0+g(x)=+ and limx0g(x)= \begin{array}{l} \lim _{x \rightarrow 0^{+}} g(x)=+\infty \text { and } \\ \lim _{x \rightarrow 0^{-}} g(x)=-\infty \end{array} \newline(C)\newlinelimx0+g(x)= and limx0g(x)=+ \begin{array}{l} \lim _{x \rightarrow 0^{+}} g(x)=-\infty \text { and } \\ \lim _{x \rightarrow 0^{-}} g(x)=+\infty \end{array} \newline(D)\newlinelimx0+g(x)= and limx0g(x)= \begin{array}{l} \lim _{x \rightarrow 0^{+}} g(x)=-\infty \text { and } \\ \lim _{x \rightarrow 0^{-}} g(x)=-\infty \end{array}
  1. Behavior of Tangent Function: Understand the behavior of the tangent function near zero.\newlineThe tangent function, tan(x)\tan(x), approaches 00 as xx approaches 00 from both the positive and negative sides. Since g(x)=1tan2(x)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.
  2. Limit as xx Approaches 00 from Positive Side: Consider the limit of g(x)g(x) as xx approaches 00 from the positive side.\newlineAs xx approaches 00 from the positive side, tan(x)\tan(x) approaches 00, and thus tan2(x)\tan^2(x) also approaches 00. 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)g(x) as xx approaches 00 from the positive side is positive infinity.\newline0044
  3. Limit as xx Approaches 00 from Negative Side: Consider the limit of g(x)g(x) as xx approaches 00 from the negative side.\newlineAs xx approaches 00 from the negative side, tan(x)\tan(x) approaches 00, and thus tan2(x)\tan^2(x) also approaches 00. 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 tan2(x)\tan^2(x) is positive whether xx approaches from the left or right, the limit from the negative side is also positive infinity.\newline0033

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