Quote by PirateFan308
I am having trouble understanding how to find the limit of a function (using the definition of a limit). I have a class example, and was wondering if anyone could walk me through the steps.
1. The problem statement, all variables and given/known data
Using the definition of the limit to show that lim_{x→2}(x^{2})=4
f(x) = x^{2}
c=2
L=4
Given an arbitrary ε>0, take δ=min{1,ε/5}
If x≠2 and x2<δ then x2<1 and x2< ε/5
f(x)L = x^{2}4 = (x2)(x+2) = x2x+2
x2<1 => 1<x<3 => 3<x+2<5 => x+2<5
x2x+2 < (ε/5)(5) = ε so f(x)L<ε
2. Relevant equations
We say that lim f(x)_{x→c}=L if:
[itex]\forall[/itex]ε>0 [itex]\exists[/itex]δ>0 [itex]\forall[/itex]x[itex]\in[/itex]dom f if x≠c and xc<δ then f(x)ε<L
3. The attempt at a solution
The biggest thing I am confused about is how the professor got δ? Did he have to do the later work first and then went back and plugged in the answer he got?
Also, in the definition, it says that then f(x)ε<L but we ended up getting f(x)L<ε. Why is this? I understand that we can rearrange the equation, but then doesn't this mess up the absolute value signs?

"The biggest thing I am confused about is how the professor got δ? Did he have to do the later work first and then went back and plugged in the answer he got?"
Your professor likely did some scratch work, starting with x^{2}4<ε, and then getting his result for δ.
"in the definition, it says that then f(x)ε<L but we ended up getting f(x)L<ε"
It should be f(x)L<ε in the definition.