Let $f$ be real function defined on (a,b). Let $p \in [a,b)$. We write $$f(p+)= \lim_{x\to p+}f(x)=q$$ if $\lim_{n\to \infty}f(p_n)=q$, for every sequence {$p_n$} in $(p,b)$ s.t $p_n \to p$. This is the definition of right hand limit of $f$ at p.
Question(1): How do I go from this limit of sequence definition to $\epsilon-\delta$ definition. My attempt: Applying theorem 4.2, without thinking too much, I get this $\lim_{x\to p}f(x)=q$. Which I known, that’s not correct.
Question(2): $\epsilon-\delta$ definition from Wikipedia. It is,$$\forall\varepsilon > 0,\exists \delta >0 s.t \forall x \in (a,b), 0 < x – a < \delta \Rightarrow |f(x) – L|<\varepsilon$$ just by looking at this definition of RHL, it is not clear to write expression like in this form $\lim_{x\to ?}f(x)=q$. Assume for a moment, that we manage to write expression in the above form. Now, can we reach to the limit of sequence definition given in theorem 4.25, for RHL?
Question(3): i can’t prove the following claim $$\lim_{x\to p}f(x)=q exist \iff f(p+)=f(p-)=q$$ using limit of sequence definition. Attempt There is nothing. After writing all definitions, there is zero progress.
Providing more context: I think $\epsilon-\delta$ definition is easier to apply than limit of sequence definition, in most problems. That is just my opinion. When I see a problem about RHL & LHL, I have to look for graphs. I can’t prove it by definition of one sided limit.
Warning: My question generally contains error and/or misunderstanding of concept. If you find something wrong then mention it on comment.
Edit: In short, I am asking a proof for two equivalent definition of RHL.
Best Answer
I was facing the same problem as you mentioned in Question (3) the last two days. I tried to tackle the if part by using contradiction, similar to the proof of the if part of Theorem 4.2. Here is the screenshot; hope it is not blured. To clarify, the $B(x,\delta) := \{t\in (a,b)\colon |x - t| < \delta\}$ is the open ball centered at $x$.