Suppose the functions f and g have the following property: for all E > 0 and all x,
if 0 < |x - 2| < sin((E^2)/9) + E, then |f(x) - 2| < E,
if 0 < |x - 2| < E^2, then |g(x) - 4| < E.
For each E > 0, find a d > 0 such that, for all x,
i) if 0 < |x - 2| < d, then |f(x) + g(x) - 6| < E.
N/A, I think.
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| < E and |g(x) - 4| < E, the best strategy seemed to be to change d so that it would produce values that would be, for each expression involving f(x) and g(x) would be less than E/2. However, since I don't actually know what f(x) and g(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.