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Here is what I have done [itex] |sin(x)| - |sin(x_0)|<|sin(x) - sin(x_0)|<\epsilon [/itex] and |sin(x)|<|x| so -|x| < -|sin(x)| => [itex] |sin(x)|- |x| < |sin(x)| - |sin(x_0)|< \epsilon [/itex] but I can’t seem to go anywhere from there.

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- Thread starter JonF
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- #1

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Here is what I have done [itex] |sin(x)| - |sin(x_0)|<|sin(x) - sin(x_0)|<\epsilon [/itex] and |sin(x)|<|x| so -|x| < -|sin(x)| => [itex] |sin(x)|- |x| < |sin(x)| - |sin(x_0)|< \epsilon [/itex] but I can’t seem to go anywhere from there.

- #2

mathman

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i tried the addition angle identity, it only seemed to make things worse :(

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HallsofIvy

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lurflurf

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1)for all real numbers x and y

sin(x/2)-sin(y/2)=2*cos(x+y)*sin(x-y)

2)for all real numbers x

cos(x)<=1

3) for all real numbers x

|sin(x)|<=|x|

Thus the given problem

show that |sin(x)-sin(y)| can be made small by chosing |x-y| small

becomes

show that |sin(x)| can be made small by chosing |x| small

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