# Epsilon Delta Limit.

by ƒ(x)
Tags: delta, epsilon, limit
 P: 323 For part A, (described here: http://www.cramster.com/solution/solution/1157440) I don't understand why they say |x-2| < 1 and why $\delta$ = min{1,ε/5} In case you can't view the page: lim x2+2x-5 = 3, x $\rightarrow$ 2 Let ε > 0 and L = 3. |x2 + 2x -5 -3| < ε |x2 + 2x - 8| < ε |x+4||x-2| < ε If |x-2| < 1, x $\rightarrow$ (1,3) $\rightarrow$ x+2 $\in$ (3,5) |x-2| < 1 $\rightarrow$ |x+2| < 5 |x-2| < 1 $\rightarrow$ |x2 +2x - 8| = |x-4||x+2| < 5|x-4| For ε > 0, 5|x-4| < ε $\rightarrow$ |x-4| < $\frac{ε}{5}$ If $\delta$ = min{1,$\frac{ε}{5}$} |x-2| < $\delta$ $\rightarrow$ |x2 + 2x - 8| < 5|x-4| < ε
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P: 26,148
hi ƒ(x)!
 Quote by ƒ(x) I don't understand why they say |x-2| < 1 and why $\delta$ = min{1,ε/5} |x+4||x-2| < ε
because if δ < 1, then |x+4| < 5

and if δ < ε/5, then |x-2| < ε/5 (obviously!)
so |x+4||x-2| < 5*ε/5 = ε
(though i don't see why they say |x-2| < 1 either )
P: 323
 Quote by tiny-tim hi ƒ(x)! because if δ < 1, then |x+4| < 5 and if δ < ε/5, then |x-2| < ε/5 (obviously!) so |x+4||x-2| < 5*ε/5 = ε (though i don't see why they say |x-2| < 1 either )
Can you explain yourself a little further? I tend to be rather slow about this kind of thing.

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P: 26,148
Epsilon Delta Limit.

 Quote by ƒ(x) Can you explain yourself a little further? I tend to be rather slow about this kind of thing.
you want the product to be < ε

one easy way to do this (there are others) is to make one of them < 1 and the other < ε

but that won't work in this case, since |4+x| isn't going to be < 1
however, it will be < 5,

so we make one of them < 5 and the other < ε/5
P: 323
 Quote by tiny-tim you want the product to be < ε one easy way to do this (there are others) is to make one of them < 1 and the other < ε but that won't work in this case, since |4+x| isn't going to be < 1 however, it will be < 5, so we make one of them < 5 and the other < ε/5
So, for a different problem, part (c) on the link, I have:

lim x^3 + 2x + 1 = 4, x --> 1

Let ε > and L = 4

|x-1| < δ, |x^3 +2x - 3| < ε

But this one doesn't factor...[EDIT] wait, yes it does (I peeked at the solution, how do I factor this by myself?).

|x-1||x^2 + x + 3| < ε
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P: 26,148
hi ƒ(x)!

(just got up …)

ok, so |x-1| < δ,

and you can obviously choose x near 4 so that |x2 + x + 3| < 33 …
so how would you finish the proof?
 Quote by ƒ(x) But this one doesn't factor...[EDIT] wait, yes it does (I peeked at the solution, how do I factor this by myself?).
you just have to guess …

in an exam, they won't give you anything difficult, so start with ±1, and work your way upwards!

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