Determine the limit l for the given a, and prove that it is the limit by showing how to find a ∂ such that |f(x)-l| < ε for all x satisfying o < |x-a| < ∂.
f(x) = x4 , arbitrary a
(Spivak's Calculus 5-3iv)
The Attempt at a Solution
l = a4
The best I can do is show that |x2 - a2| < ε for |x-a|<∂1 = min(ε/(2|a|+1),1). After that, I get lost.
Since I found that |x2 - a2| < ε for |x-a|<∂1 = min(ε/(2|a|+1),1), can I rearrange |x4 - a4| to |(x2)2 - (a2)2| to find that ∂2=min(1,ε/(2|a|2+1)? This seems to be what Spivak's answer book is suggesting, but I'm not sure.
Can someone walk me through this? I'm pretty new to delta-epsilon proofs.