Claim: $\lim_{x\to 1} \frac{100}{x} = 100$
Proof: Let any $\epsilon > 0.\ |\frac{100}{x}-100| = 100 |\frac{1}{x}-1| = 100 \frac{|x-1|}{|x|}.$
If this term is smaller than epsilon in the interval where $0<|x-1|<\delta$, then we are done.
Assume $\delta=100.$ Then, $|x-1|<100, -99
Thus $100 \frac{|x-1|}{|x|} <\frac{ 100}{100*|x-1|}$
So if $0<|x-1|< \min\{\epsilon, 100\}=\delta,$ then $|\frac{100}{x}-100| < \epsilon$. QED
Note: Even if this is correct, my intent of asking this question was because I'm having doubts on the fundamental logic lying behind this proof... For some reason, I don't like the fact that we can let delta be something and then later "rectify" it after putting in a form of epsilon... Like I can let delta be less than 1/2 and get (I think) for delta = min{epsilon/200, 1/2} and that's different from what I got... I know that epsilon is any number, but it just seems weird. Did I misunderstand something?
I will be really grateful if someone can help me in any way understand this better.