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Re: Limit Problem
Posted:
Jan 26, 2013 7:35 PM


On Jan 26, 10:56 pm, "Charles Hottel" <chot...@earthlink.net> wrote: > I am having a problem following an example in my book. > I understand the concept of limit but sometimes I get confused > manipulating expressions with absolute values in them. Here is the problem: > > Prove lim(x>c) 1/x = 1/c, c not equal zero > > So 0 <  xc < delta, implies 1/x  1/c < epsilon > > 1/x  1/c =  (cx) / {xc} = 1/x * 1/c * (xc) < epsilon > > Factor 1/x is troublesome if x is near zero, so we bound it to keep it > away from zero. > > So c = c  x + x <= cx + x and this imples x >= c  xc > > I think I understand everything up to this point, but not the next steps, > which are > > If we choose delta <= c/2 we succeed in making x >= c / 2. > Finally if we require delta <= [(epsilon) * (c**2)} / 2 then > > [1/x * 1/c * xc] < [1 / (c/2)] * [1/c] * [((epsilon) * > (c**2)) / 2] = epsilon > > How did they know to choose delta <= c/2? > > How does that lead to x > c/2 implies 1/x < 1/(c/2) ? > > I did not sleep well last night and I feel I must be missing something > that would be obvious if my head was clearer. Thanks for any help.
"If we choose delta <= c/2 we succeed in making x >= c / 2. "
Once delta is chosen, you will need to examine all x with x  c < delta and show that 1/x is close to 1/c.
You want to choose delta so that x stays far away from zero.
Let's say you picked delta = 0.01. If for example c = 1.43, then delta = 0.01 means x  1.43 < 0.01, or 1.42 < x < 1.44. That's far away from x = 0. But the same delta with c = 0.012 wouldn't be very good, because delta = 0.01 now means that x  0.012 < 0.01, or 0.002 < x < 0.022. If c = 0.005 then this delta is awful, because x  0.005 < 0.01 means 0.005 < x < 0.015; even x = 0 would be included.
What happens if we pick delta = c / 1000? It means that x has to be very, very close to c. Actually, x must be between 0.999c and 1.001c. We don't know if c is positive or negative, so we can't just say 0.999c < x < 1.001 c because that would be wrong if c is negative. But we can say that 0.999 c < x < 1.001 c. Similar, if we pick delta = c / 2 then x must be between 0.5 c and 1.5 c, or 0.5 c < x < 1.5 c.



