New to calc, need help with limits plz

[slinky2004]

basically, we're supposed to be finding delta(d) given epsilon(E) or proving a limit using the definition of a limit:

i can do it if everything works out simply like:

f(x)=x+1. Find d such that if 0<|x-2|<d then |f(x)-3|<.04

i do:
|x+1-3|<.04
1|x-2|<.04
so d=.04

but something like this throws me off:

f(x)=1/(x-1). Find d such that if 0<|x-2|<d then |f(x)-1|<.01

i cant figure out how to factor 1/(x-1)-1 so that its in the format coefficent*|x-2|.

Here's one of the proofs that i cant get:

prove using the epsilon-delta definition:
lim(2x+5)=-1
x-->-3

i do:
|2x+4|<E where 0<|x+3|<d

and again, i cant figure out how to factor 2x+4 so the format coefficent*|x-2|.

 

[stapel]

Usually it helps to work things out, and figure out at the end what you needed. Then you get a fresh piece of paper, and work backwards, making it look like magic (ya know, like it is in the books).

If you can do f(x) = x + 1, you shouldn't have any difficulty with g(x) = 2x + 5, since it's the same type of function. But h(x) = 1/(x - 1) is a little different. Let's look at a related problem:

. . . .Let f(x) = 1/x. Show that the limit of f(x)
. . . .as x approaches 1 is 1.

. . . .Let's see what we get...

. . . .|1/x - 1/1| = |(1 - x)/x| < e
. . . .-e < (1 - x)/x < e

. . . .For x really close to 1, 1 - x and x are
. . . .both positive, so:

. . . .0 < (1 - x)/x < e

. . . .Now if I've taken x so |1 - x| < d, then:

. . . .0 < (1 - x)/x < d/x < e (I hope!)

. . . .-d < 1 - x < d
. . . .d > x - 1 > -d
. . . .d + 1 > x > 1 - d

. . . .If you divide by something bigger,
. . . .you get something smaller, so:

. . . .d/x > d/(d + 1)

. . . .Then:

. . . .0 < d/(d + 1) < d/x < e (I hope!)

. . . .So, for whatever epsilon e I'm given,
. . . .I need to pick delta d so d/(d + 1) < e.
. . . .Since e is going to be very close to zero,
. . . .then e - 1 is going to be negative, so:

. . . .d/(d + 1) < e
. . . .d < ed + e
. . . .-e < ed - d
. . . .-e < d(e - 1)
. . . .-e/(e - 1) > d
. . . .e/(1 - e) > d

. . . .So, for any given e, I need to pick d so
. . . .d < e/(1 - e). Now I can do my proof....

And so forth.

Eliz.