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Let 
f(x)=(-x)/(ln^(2)(x-1)).
Select the correct description of the one-sided limits of 
f at 
x=2.
Choose 1 answer:
(A) 
lim_(x rarr2^(+))f(x)=+oo and 
lim_(x rarr2^(-))f(x)=+oo
(B) 
lim_(x rarr2^(+))f(x)=+oo and 
lim_(x rarr2^(-))f(x)=-oo
(C) 
lim_(x rarr2^(+))f(x)=-oo and 
lim_(x rarr2^(-))f(x)=+oo
(D) 
lim_(x rarr2^(+))f(x)=-oo and 
lim_(x rarr2^(-))f(x)=-oo

Let f(x)=xln2(x1) f(x)=\frac{-x}{\ln ^{2}(x-1)} .\newlineSelect the correct description of the one-sided limits of f f at x=2 x=2 .\newlineChoose 11 answer:\newline(A) limx2+f(x)=+ \lim _{x \rightarrow 2^{+}} f(x)=+\infty and limx2f(x)=+ \lim _{x \rightarrow 2^{-}} f(x)=+\infty \newline(B) limx2+f(x)=+ \lim _{x \rightarrow 2^{+}} f(x)=+\infty and limx2f(x)= \lim _{x \rightarrow 2^{-}} f(x)=-\infty \newline(C) limx2+f(x)= \lim _{x \rightarrow 2^{+}} f(x)=-\infty and limx2f(x)=+ \lim _{x \rightarrow 2^{-}} f(x)=+\infty \newline(D) limx2+f(x)= \lim _{x \rightarrow 2^{+}} f(x)=-\infty and limx2f(x)= \lim _{x \rightarrow 2^{-}} f(x)=-\infty

Full solution

Q. Let f(x)=xln2(x1) f(x)=\frac{-x}{\ln ^{2}(x-1)} .\newlineSelect the correct description of the one-sided limits of f f at x=2 x=2 .\newlineChoose 11 answer:\newline(A) limx2+f(x)=+ \lim _{x \rightarrow 2^{+}} f(x)=+\infty and limx2f(x)=+ \lim _{x \rightarrow 2^{-}} f(x)=+\infty \newline(B) limx2+f(x)=+ \lim _{x \rightarrow 2^{+}} f(x)=+\infty and limx2f(x)= \lim _{x \rightarrow 2^{-}} f(x)=-\infty \newline(C) limx2+f(x)= \lim _{x \rightarrow 2^{+}} f(x)=-\infty and limx2f(x)=+ \lim _{x \rightarrow 2^{-}} f(x)=+\infty \newline(D) limx2+f(x)= \lim _{x \rightarrow 2^{+}} f(x)=-\infty and limx2f(x)= \lim _{x \rightarrow 2^{-}} f(x)=-\infty
  1. Analyze near x=2x = 2: Analyze the function near x=2x = 2. We have the function f(x)=xln2(x1)f(x) = \frac{-x}{\ln^{2}(x-1)}. We need to consider the behavior of the function as xx approaches 22 from the left (x2x \to 2^{-}) and from the right (x2+x \to 2^{+}). We will look at the numerator and the denominator separately.
  2. Numerator behavior at x=2x = 2: Consider the behavior of the numerator as xx approaches 22. The numerator is x-x. As xx approaches 22 from either side, x-x approaches 2-2. There is no ambiguity or complexity in the behavior of the numerator.
  3. Denominator behavior at x=2x = 2: Consider the behavior of the denominator as xx approaches 22. The denominator is ln2(x1)\ln^{2}(x-1), which is the square of the natural logarithm of (x1)(x-1). As xx approaches 22, (x1)(x-1) approaches 11. The natural logarithm of 11 is xx00, so ln2(x1)\ln^{2}(x-1) approaches xx00. We need to consider the sign of the natural logarithm as xx approaches 22 from the left and right to determine the sign of the denominator.
  4. Denominator sign at x2x \to 2^{-}: Determine the sign of the denominator as xx approaches 22 from the left. As xx approaches 22 from the left (x2x \to 2^{-}), (x1)(x-1) is slightly less than 11. The natural logarithm of a number slightly less than 11 is negative, so ln(x1)\ln(x-1) is negative. Squaring a negative number gives a positive result, so xx00 is positive as xx approaches 22 from the left.
  5. Denominator sign at x2+x \to 2^{+}: Determine the sign of the denominator as xx approaches 22 from the right.\newlineAs xx approaches 22 from the right (x2+x \to 2^{+}), (x1)(x-1) is slightly more than 11. The natural logarithm of a number slightly more than 11 is positive, so ln(x1)\ln(x-1) is positive. Squaring a positive number gives a positive result, so xx00 is positive as xx approaches 22 from the right.
  6. Combine for one-sided limits: Combine the behavior of the numerator and denominator to find the one-sided limits.\newlineSince the numerator approaches 2-2 and the denominator approaches 00 with a positive sign from both sides, the fraction xln2(x1)\frac{-x}{\ln^{2}(x-1)} will approach negative infinity from both sides. This is because a negative number divided by a positive number that is getting closer and closer to zero will result in a number that is getting more and more negative.

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