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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)=1(x1)2 f(x)=-\frac{1}{(x-1)^{2}} .\newlineSelect the correct description of the one-sided limits of f f at x=1 x=1 .\newlineChoose 11 answer:\newline(A)\newlinelimx1+f(x)=+ and limx1f(x)=+ \begin{array}{l} \lim _{x \rightarrow 1^{+}} f(x)=+\infty \text { and } \\ \lim _{x \rightarrow 1^{-}} f(x)=+\infty \end{array} \newline(B)\newlinelimx1+f(x)=+ and limx1f(x)= \begin{array}{l} \lim _{x \rightarrow 1^{+}} f(x)=+\infty \text { and } \\ \lim _{x \rightarrow 1^{-}} f(x)=-\infty \end{array} \newline(c)\newlinelimx1+f(x)= and limx1f(x)=+ \begin{array}{l} \lim _{x \rightarrow 1^{+}} f(x)=-\infty \text { and } \\ \lim _{x \rightarrow 1^{-}} f(x)=+\infty \end{array} \newline(D)\newlinelimx1+f(x)= and limx1f(x)= \begin{array}{l} \lim _{x \rightarrow 1^{+}} f(x)=-\infty \text { and } \\ \lim _{x \rightarrow 1^{-}} f(x)=-\infty \end{array}

Full solution

Q. Let f(x)=1(x1)2 f(x)=-\frac{1}{(x-1)^{2}} .\newlineSelect the correct description of the one-sided limits of f f at x=1 x=1 .\newlineChoose 11 answer:\newline(A)\newlinelimx1+f(x)=+ and limx1f(x)=+ \begin{array}{l} \lim _{x \rightarrow 1^{+}} f(x)=+\infty \text { and } \\ \lim _{x \rightarrow 1^{-}} f(x)=+\infty \end{array} \newline(B)\newlinelimx1+f(x)=+ and limx1f(x)= \begin{array}{l} \lim _{x \rightarrow 1^{+}} f(x)=+\infty \text { and } \\ \lim _{x \rightarrow 1^{-}} f(x)=-\infty \end{array} \newline(c)\newlinelimx1+f(x)= and limx1f(x)=+ \begin{array}{l} \lim _{x \rightarrow 1^{+}} f(x)=-\infty \text { and } \\ \lim _{x \rightarrow 1^{-}} f(x)=+\infty \end{array} \newline(D)\newlinelimx1+f(x)= and limx1f(x)= \begin{array}{l} \lim _{x \rightarrow 1^{+}} f(x)=-\infty \text { and } \\ \lim _{x \rightarrow 1^{-}} f(x)=-\infty \end{array}
  1. Understand function behavior: Analyze the function f(x)=1(x1)2f(x) = -\frac{1}{(x-1)^{2}} to understand its behavior near x=1x = 1. The function is a rational function with a negative sign in front and a squared denominator. This means that as xx approaches 11, the denominator approaches 00, 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).
  2. Calculate right-hand limit: Calculate the right-hand limit as xx approaches 11 from the right (x1+x \to 1^+).\newlineAs xx gets closer to 11 from the right, (x1)(x - 1) becomes a small positive number. Squaring a small positive number gives a small positive number. Dividing 1-1 by a small positive number gives a large negative number. Therefore, the right-hand limit is negative infinity.\newlinelimx1+f(x)=\lim_{x \to 1^+} f(x) = -\infty
  3. Calculate left-hand limit: Calculate the left-hand limit as xx approaches 11 from the left (x1x \to 1^-).\newlineAs xx gets closer to 11 from the left, (x1)(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 1-1 by a small positive number again gives a large negative number. Therefore, the left-hand limit is also negative infinity.\newlinelimx1f(x)=\lim_{x \to 1^-} f(x) = -\infty
  4. Describe one-sided limits: Combine the results from Step 22 and Step 33 to describe the one-sided limits of ff at x=1x = 1. Both the right-hand and left-hand limits of ff as xx approaches 11 are negative infinity. Therefore, the correct description of the one-sided limits of ff at x=1x = 1 is: limx1+f(x)=\lim_{x \to 1^+} f(x) = -\infty and limx1f(x)=\lim_{x \to 1^-} f(x) = -\infty

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