# Thread: finding limits graphically and numerically

1. ## finding limits graphically and numerically

I understand everything but the part in red. how do you get delta = E/3?

2. ## I think they mean "take delta = 1/3"

On the delta-epsilon formulation, delta is a free variable and you've to prove that for any positive delta you can find an epsilon such that abs(f(x) - limit) < epsilon. You could as well take delta = 1/2 or .0001.

3. My bad - read 1/3 in place of epsilon/3.
In any case I guess they mean you can take espilon = 3 * delta.
Point is you can choose any positive delta and that determines epsilon.

4. Originally Posted by algebra2
I understand everything but the part in red. how do you get delta = E/3?

in fact, $\displaystyle \delta = \frac {\epsilon}2$ would work i think

anyway, recall what the definition of a limit means.

we want to find a $\displaystyle \delta > 0$, such that for all $\displaystyle \epsilon > 0$ (and $\displaystyle x \in \text{dom}(f)$), $\displaystyle |x - 1| < \delta$ implies $\displaystyle |f(x) - 2| < \epsilon$.

now, we found that $\displaystyle |x - 1| < \frac {\epsilon}{x + 1}$, but we want $\displaystyle |x - 1|< \delta$, so we need to find a $\displaystyle \delta$ that works. now, what does it mean for $\displaystyle x$ to be close to 1? we give ourselves a generous range. and say, let it be somewhere between 0 and 2, and those x's are "close". now, to make sure our $\displaystyle \delta$ works, we decide to be cautious and choose $\displaystyle x = 2$, that way, $\displaystyle x$ gets really close to 1, since $\displaystyle \frac {\epsilon}{x + 1}$ is the smallest for the range we are considering. so, if $\displaystyle x = 2$, we have $\displaystyle \frac {\epsilon}{x + 1} = \frac {\epsilon}3$, and so we choose that as our $\displaystyle \delta$.

ok, so that was hopelessly confusing, even to me, i know what i want to say, but i am not sure if i said it ok. did you get that?

Originally Posted by hwhelper
My bad - read 1/3 in place of epsilon/3.
In any case I guess they mean you can take espilon = 3 * delta.
Point is you can choose any positive delta and that determines epsilon.
actually, it is delta that depends on epsilon, not the other way around. we must choose a delta that works for any epsilon we are given

5. yeah, thanks for the explanation

6. Originally Posted by algebra2
yeah, thanks for the explanation
good, i was worried. i hope you realize that things like $\displaystyle |x - 1|< \delta$ is talking about distance. the distance between x and 1 is less than delta, that's what it means. this is why i was talking about x "close" to 1, etc

7. See my signature for more details.

8. Originally Posted by Krizalid
See my signature for more details.
your signature is so functional!