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 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 limx→2(x2)=4 f(x) = x2 c=2 L=4 Given an arbitrary ε>0, take δ=min{1,ε/5} If x≠2 and |x-2|<δ then |x-2|<1 and |x-2|< ε/5 |f(x)-L| = |x2-4| = |(x-2)(x+2)| = |x-2||x+2| |x-2|<1 => 1 3 |x+2|<5 |x-2||x+2| < (ε/5)(5) = ε so |f(x)-L|<ε 2. Relevant equations We say that lim f(x)x→c=L if: $\forall$ε>0 $\exists$δ>0 $\forall$x$\in$dom f if x≠c and |x-c|<δ then |f(x)-ε|
"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 |x2-4|<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.