Okay so I have this Comparative Economic Systems final in 13 days. It’s 25% of my grade, and cumulative. I basically have to know everything about capitalism, socialism, and the economic framework and history of 7 countries to the point where I could compare and contrast any aspect of them with any other country.
So instead of starting to study the insurmountable amount of information I’m required to know, or finding more clues for the scavenger hunt, I’m just going to prove that thing we talked about in class on Monday.
Using the definition of limit given, prove:
lim (x→0) x² = 0
lim(x→17) x² = 17²
The definition of limit as given before in class is: (∀ε>0)(∃δ>0) such that if 0<|x-a|<δ then |f(x) – L|<ε.
We’ll start with lim (x→0) x² = 0. First, let’s define some terms/match them up with the definition of a limit.
f(x) = x²
a = 0
L = 0.
So based on the definition, if 0<|x|<δ, then |x²|<ε. Since the definition says that this definition must work for all ε>0, we just need to choose one δ that makes this statement true. If δ ≥0, this statement works, since x is approaching 0.
0<|x|<1, then |0-0|<ε, which is true because ε>0.
For lim(x→17) x² = 17²,
f(x) = x²
a = 17
L = 17².
Based on the definition, if 0<|x-17|<δ then |x² – 17² |<ε.
Adding 17 to both sides, we get that if -17<x<17, then |x² – 17² |<ε, which is true, because ε must be greater than 0. No matter what you put in for x, because of the absolute values, cannot be less than 0.
That’s really all I’ve got. I thought these would be much easier to prove since we’re given a definition to plug things into, but I don’t think I proved either of these. I’m ignoring the fact that I need to assume the hypothesis is true and focus on the conclusion, and I’ve lost sight of what needs to really be proved in this statement, since we’ve got δ and ε to prove.
Dang it. And I thought I was being productive.