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$Let f(x)=-(1)/((x-1)^(2)). Select the correct description of the one-sided limits of f at x=1. Choose 1 answer: (A) {:[lim_(x rarr1^(+))f(x)=+oo" and "],[lim_(x rarr1^(-))f(x)=+oo]:} (B) {:[lim_(x rarr1^(+))f(x)=+oo" and "],[lim_(x rarr1^(-))f(x)=-oo]:} (c) {:[lim_(x rarr1^(+))f(x)=-oo" and "],[lim_(x rarr1^(-))f(x)=+oo]:} (D) {:[lim_(x rarr1^(+))f(x)=-oo" and "],[lim_(x rarr1^(-))f(x)=-oo]:}$

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

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Math Problems

Algebra 1

Compare linear, exponential, and quadratic growth

Full solution

Q. Let

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

\newline

Select the correct description of the one-sided limits of

f

x=1

\newline

Choose

1

answer:

\newline

(A)

\newline

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

\newline

(B)

\newline

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

\newline

(c)

\newline

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

\newline

(D)

\newline

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

Understand function behavior: Analyze the function $f(x) = -\frac{1}{(x-1)^{2}}$ to understand its behavior near $x = 1$ . The function is a rational function with a negative sign in front and a squared denominator. This means that as $x$ approaches $1$ , the denominator approaches $0$ , which will cause the value of the function to approach infinity. However, the negative sign will affect the direction of the infinity (positive or negative).
Calculate right-hand limit: Calculate the right-hand limit as $x$ approaches $1$ from the right ( $x \to 1^+$ ). $\newline$ As $x$ gets closer to $1$ from the right, $(x - 1)$ becomes a small positive number. Squaring a small positive number gives a small positive number. Dividing $-1$ by a small positive number gives a large negative number. Therefore, the right-hand limit is negative infinity. $\newline$ $\lim_{x \to 1^+} f(x) = -\infty$
Calculate left-hand limit: Calculate the left-hand limit as $x$ approaches $1$ from the left ( $x \to 1^-$ ). $\newline$ As $x$ gets closer to $1$ from the left, $(x - 1)$ becomes a small negative number. Squaring a small negative number still gives a small positive number (since the square of any real number is non-negative). Dividing $-1$ by a small positive number again gives a large negative number. Therefore, the left-hand limit is also negative infinity. $\newline$ $\lim_{x \to 1^-} f(x) = -\infty$
Describe one-sided limits: Combine the results from Step $2$ and Step $3$ to describe the one-sided limits of $f$ at $x = 1$ . Both the right-hand and left-hand limits of $f$ as $x$ approaches $1$ are negative infinity. Therefore, the correct description of the one-sided limits of $f$ at $x = 1$ is: $\lim_{x \to 1^+} f(x) = -\infty$ and $\lim_{x \to 1^-} f(x) = -\infty$