Epsilon - delta

Could you please elaborate your answer. Honestly curious.

t. op

>ive never see anyone be able to do this without having to resort to infinite series
It's a demonstration of the following theorem, which is very easily proved using epsilon-delta: let [math]a_n[/math] be bounded and [math]b_n[/math] such that [math]\lim_{n \to \infty} b_n = 0[/math]. Then [math]\lim_{n\to \infty}a_n b_n = 0[/math].

When you translate calculus into languages outside of latin roots, it is actually a form of algebra.

Just because the definition is introduced in calc 1 doesn't mean it's full utility is realized there, calc 1 is not in sufficient generality. You need the definition to prove theorems that show that the things you do in calculus work. But you won't really see this until analysis. As far as the sin x over x limit, sine is, intensionally speaking, in analysis an infinite series. Thus to prove things about it you must use series. Most analytic machinery is developed without have sine. So trigonometic identities might help with simple limits, but as problems get more complicated, more machinery is necessary. Another example is the pull back of open sets is open definition of continuity. Sometimes that definition is much easier, and even necessary, to use to prove things.

What do you mean? We're having a thread about limits. If someone knows something about them they'd be perfect contributors. This guy seems to have a strong opinion that e-d limits are useful. I'm sure we could learn something if he told us how.

This whole board is going to become useless if people keep coming in to threads specifically to let people know they're wrong - as if asking a naive question makes you not even worth the time of day. Meanwhile, online IQ test threads and /x/-worthy discussions about spatial dimensions flourish.

Ahh, now that is helpful.

OP don't listen to brainlets saying that it's useless. It's absolutely necessary to make sense of limiting processes. Without epsilon-delta reasoning, everything involving limits (derivatives, integration, infinite series) become just handwaving. Keep in mind, that all exercises that you do in calculus are artificially made up so that "everything works nicely".

good post

useless. modern world is built on physics, not math proof circlejerking, and guess what physicists use: infinitesimals.

It says that, given [math] \text{any} [/math] positive real number [math] \epsilon [/math], you can ensure that [math] f(x) [/math] will [math] \text{only} [/math] take on values in the [math] \epsilon [/math]-neighborhood of [math] L_c [/math] (We're assuming the space is Hausdorff, so L_c will be unique) if you restrict the values [math] x [/math] can take to a [math] \delta [/math]-neighborhood of [math] c [/math].

In case [math] f [/math] is given to be continuous on a set [math] S [/math], i.e. [math] L_c = f(c) [/math] for all [math] c [/math] in [math] S [/math], observe that such a [math] \delta [/math] will depend not only on the chosen [math] \epsilon [/math] but also on [math] c [/math] and [math] S [/math]. It will occasionally be useful if this dependency of [math] \delta [/math] on the point [math] c [/math] could be ignored. If that is indeed the case, [math] f [/math] is said to be uniformly continuous on S. It would be awkward to define or understand useful properties such as uniform continuity without precise definitions of limits.

Thanks guise.