This one is a simple "plug-and-chug," as they say: simply plug in 2 for "x," and see what happens. If you really want to, you can plug in 10 for x as well, just to make sure you're on the right track.For Quantity A, if you plug in a 2, then you get:1/(1-(1/(1+(1/2))))=1/(1-(1/(3/2)))= 1/(1-(2/3))=1/(1/3)= 3For Quantity B:1/(1+(1/(1-(1/2))))= 1/(1+(1/(1/2)))= 1/(1+2)= 1/3Obviously, A is greater than B.Just to be sure, plug in a 10 for x. In that case,Quantity A = 11Quantity B = 9/19At this point, it's obvious that, as x gets larger, Quantity B remains a fraction less than 1, whereas Quantity A keeps increasing as x increases. A will always be greater than B.For fun, I tried plugging in 10^6 for x. The result was still in favor of Quantity A. If you were to graph Quantity B as a function, you'd see that, as x increases in value, the function approaches 1/2 but can never reach it: 1/2 is, in fact, an asymptote! We can prove this by setting Quantity B equal to 1/2. If you do the math, you get the absurd result that -1 = 0. Such absurdities alert us to the presence of an asymptote. So: f(x) can never equal or surpass 1/2. f(x) < 1/2 as x approaches ∞.
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