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so I was wondering if there is any part of the $\varepsilon$-$\delta$ definition of the limit that offers any insight on how to find the limit of a function, or if this is something you are supposed to guess at based on the function itself or other elementary functions whose limits you have found.

For example, if I have the simple function

$$\lim_{x\rightarrow 0} \rvert x \lvert$$

Can I use the epsilon delta definition to figure out the limit? Or do I just have to make a reasonable guess that it is 0 and then use the epsilon delta definition to prove it?

Thanks

Michael Hardy
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user1236
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  • Not necessarily. The $\varepsilon-\delta$ definition does however introduce Cauchy sequences. With this we may numerically calculate a sequence to the $n^{th}$ term knowing that it will be fairly close to its limit. However, this would only be useful for a numerical approximation. The only true way to find a limit is analytically. – Eoin Oct 29 '14 at 03:45
  • Generally no, you'll need to guess the value of the limit. Fortunately in practice this is often not too difficult. – Gyu Eun Lee Oct 29 '14 at 03:46

3 Answers3

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In computing limits, writing something like $\lim\limits_{x\to3}\dfrac{x-3}{x^2-9}$, one uses algebra to reduce the problem to $\lim\limits_{x\to3}\dfrac{1}{x+3}$, one uses continuity of the reciprocal function and the polynomial function in the denominator to justify plugging in $3$ at that point. The point at which $\varepsilon$-$\delta$ proofs enter the process would be in proving that things like polynomial functions and the reciprocal function are continuous.

Michael Hardy
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The $\epsilon$-$\delta$ definition will not tell you how to find a limit, because it depends on already knowing the limit. That is, say we have a function $f(x)$. Then, $\lim\limits_{x\to p} f(x) = P~$ if...

$$\forall\epsilon > 0. ~\exists \delta > 0 \ni \forall x \in\text{Dom}(f).~ |x - p| < \delta \implies |f(x) - P | < \epsilon$$ To put it in more easily understood terms: The $\epsilon$-$\delta$ definition of a limit says that $P$ is a limit of $f(x)$ if and only if $f(x)$ gets closer to $P$ as $x$ gets closer to $p$ (the $\epsilon$ and $\delta$ here are like measurements of distance). This definition only gives us a criterion for figuring out whether or not $P$ is a limit—but we have to first know $P$ in order to use the definition.

Theoretically, you could use the $\epsilon$-$\delta$ definition to infer a limit, but that'd be the equivalent of just looking at $f(x)$ itself (with some extra, explicit parameters that may make things more difficult).

AmagicalFishy
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According my knowledge, language is only used to prove that the limit is, for example, A, however, it is not used to compute what is the limit.

Paul
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