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

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

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Math Problems

Algebra 1

Compare linear, exponential, and quadratic growth

Full solution

Q. Let

h(x)=-\frac{2 x}{(x+1)^{2}}

\newline

Select the correct description of the one-sided limits of

h

x=-1

\newline

Choose

1

answer:

\newline

(A)

\newline

\begin{array}{l} \lim _{x \rightarrow-1^{+}} h(x)=+\infty \text { and } \\ \lim _{x \rightarrow-1^{-}} h(x)=+\infty \end{array}

\newline

(B)

\newline

\begin{array}{l} \lim _{x \rightarrow-1^{+}} h(x)=+\infty \text { and } \\ \lim _{x \rightarrow-1^{-}} h(x)=-\infty \end{array}

\newline

(C)

\newline

\begin{array}{l} \lim _{x \rightarrow-1^{+}} h(x)=-\infty \text { and } \\ \lim _{x \rightarrow-1^{-}} h(x)=+\infty \end{array}

\newline

(D)

\newline

\begin{array}{l} \lim _{x \rightarrow-1^{+}} h(x)=-\infty \text { and } \\ \lim _{x \rightarrow-1^{-}} h(x)=-\infty \end{array}

Analyze Function Near $-1$ : Analyze the function $h(x)$ near $x = -1$ . The function is given by $h(x) = -\frac{2x}{(x+1)^{2}}$ . We need to find the one-sided limits as $x$ approaches $-1$ from the right ( $x \to -1^+$ ) and from the left ( $x \to -1^-$ ).
Limit from Right: Find the limit as $x$ approaches $-1$ from the right ( $x \to -1^+$ ). $\newline$ As $x$ approaches $-1$ from the right, the numerator $-2x$ approaches $2$ , while the denominator $(x+1)^2$ approaches $0$ . Since the denominator is squared, it will always be positive, and as $x$ approaches $-1$ from the right, the denominator approaches $0$ through positive values. Therefore, the function $-1$ $2$ approaches negative infinity.
Limit from Left: Find the limit as $x$ approaches $-1$ from the left ( $x \to -1^-$ ). $\newline$ As $x$ approaches $-1$ from the left, the numerator $-2x$ again approaches $2$ , while the denominator $(x+1)^2$ still approaches $0$ . Similar to the right-hand limit, the denominator is squared and will always be positive, even as $x$ approaches $-1$ from the left. Thus, the function $-1$ $1$ also approaches negative infinity from the left.
Combine Results: Combine the results from Step $2$ and Step $3$ to determine the one-sided limits. $\newline$ From Step $2$ , we have $\lim_{x \to -1^+} h(x) = -\infty$ . $\newline$ From Step $3$ , we have $\lim_{x \to -1^-} h(x) = -\infty$ . $\newline$ Therefore, both one-sided limits as $x$ approaches $-1$ are negative infinity.