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anonymous
 one year ago
if lim(x>0)f(x)=1find the lim(x>0) (f(x)+1)sin(pi/x^2)
anonymous
 one year ago
if lim(x>0)f(x)=1find the lim(x>0) (f(x)+1)sin(pi/x^2)

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jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1Could we use the squeeze theorem on this one?

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1Consider that \[(f(x) + 1) \cdot 1 \leq (f(x) + 1) \cdot \sin(\pi/x^2) \leq (f(x) + 1) \cdot 1\] hence, \[\lim_{x\rightarrow0}(f(x) + 1) \cdot 1 \leq \lim_{x\rightarrow0}(f(x) + 1) \cdot \sin(\pi/x^2) \leq \lim_{x\rightarrow0}(f(x) + 1) \cdot 1\] that is, \[ 0\leq \lim_{x\rightarrow0}(f(x)+1) \cdot \sin(\pi/x^2) \leq 0\] and our desired limit is simply 0.

jtvatsim
 one year ago
Best ResponseYou've already chosen the best response.1The sin(pi/x^2) feels uncomfortable since we "know" that the inside is heading toward something undefined (or infinite), but the sin function itself is always stuck between 1 and 1 no matter what the inside is.
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