- #1

WhyNerfZed

- 3

- 0

I'm new to limits and calculus in general. Our professor told us there existed some rigorous proof for a limit, but it was "beyond the scope of the course". All we needed to know about a limit was that

**(1)**$$\lim_{x\to a} f(x)$$ is true iff when x approaches a from both directions p(x) approaches L. But i hate using something without being able to properly define it, so i looked up the epsilon-delta definition of a limit. After looking at it for some time, I think I am close to understanding it, but there are still something about it that just don't click in my head(it's so abstract to me). In this thread I'm going try lay out my understanding of the epsilon delta definition, and how it is equivalent to the f(x) → L when x → a^ definition. I hope you can correct me if/when I'm wrong, and help me see how, intuitively, the epsilon-delta definition says the same thing as the other definition. So here goes.

My first issue is how i decipher the general limit expression $$\lim_{x\to a} f(x)$$

So i will try to understand the one sided limit instead $$\lim_{x\to a^-} f(x)$$

How i see it, is that as x approaches a(from the negative direction), f(x) will approach L incrementally. Which implies that that no matter how close L-ε you want to come to L, there exists some δ such that f(a-δ) will evaluate to precisely L-ε. And since as x gets closer to a, f(x) gets closer to L, it must mean that for every ε there exists some x where x > a-δ Λ x < a will evaluate to function values within <L-ε, L>, or in other words L-F(x)< ε. So if all that i have said is true, then if f(x) approaches L when x approaches a, it must mean that ∀ε ∃δ such that if (a-x)<δ ⇒ (L-f(x))< ε ⇔ f(x)→ L when x→a

In essence, what I'm struggling with is how to rephrase the f(x) → L when x → a^

^{-}definition that i understand, to the epsilon delta definition.

I can see that (f(x) → L when x → a^

^{-}) ⇒ any y=L-ε can be achieved by finding an x sufficiently close to a. However the two statements are not equivalent because the last statement does not imply that f(x) → L when x → a^

^{-}. Because it is possible that x<z<a and f(x) is within ε but f(y) is not. So in order for f(x) → L when x → a^

^{-}and the limit to exists , there must exists some x such that, no matter how close ε you want to get to L, f(x) will be that close and every point beyond

**x**and up to

**a**will evaluate to functions as close or closer to

**a**.

Sorry for the wall of text with few, if any, direct questions. Would greatly appreciate any corrections to my post or pointers to how i can understand this better. I can't seem to move on with my school work until i have internalized this definition.