(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose the functionsfandghave the following property: for allE > 0and allx,

if0 < |x - 2| < sin((E^2)/9) + E, then|f(x) - 2| < E,

if0 < |x - 2| < E^2, then|g(x) - 4| < E.

For eachE > 0, find ad > 0such that, for allx,

i) if0 < |x - 2| < d, then|f(x) + g(x) - 6| < E.

2. Relevant equations

N/A, I think.

3. The attempt at a solution

Well, what I did was look at|f(x) + g(x) - 6| < E. Since I was given|f(x) - 2| < Eand|g(x) - 4| < E, the best strategy seemed to be to changedso that it would produce values that would be, for each expression involvingf(x)andg(x)would be less thanE/2. However, since I don't actually know whatf(x)andg(x)are, I'm at a loss as to how to do that.

Spivak's solution (since this problem comes from there, ch. 5 #6), says the same thing ("we need...< E/2") but then says that this means I need:

0 < |x - 2| < min(sin(E^2/36)^2 + E/2, E^2/4) = d

...the logic of which escapes me.

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# Homework Help: Delta-Epsilon Proofs

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