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Is there any limit like this

$$\lim_{f(x)\to0} g(x)=0?$$

Defined as follows$$\forall \epsilon>0,\exists \delta>0 | |g(x)|<\epsilon ,\forall x |0<|f(x)|<\delta?$$ Where f and g real functions, such that range of f may not contain $0$.

In the real analysis book by bartle page - 201 (PDF page-217), in the remark following Riemann integral part , it says that the Riemann integral as defined in the text , is often called limit tends to infinity , but the author mentions this limit is not the usual limit as discussed earlier in the book.

So my question is can we define the limit more generally as in above and what are the properties(example additive,multiplicative etc) of this limit, that makes it different from normal limit (where informally the g(x) is an explicit function of the tending variable) ?

Are the following statements true or false? (all the functions are assumed to be real functions and the limits are assumed to exist)

$$\lim_{f(x)\to0} (g(x)+p(x))=\lim_{f(x)\to0} g(x)+\lim_{f(x)\to0} p(x)$$ $$\lim_{f(x)\to0} (g(x)\cdot p(x))=\lim_{f(x)\to0} g(x)\cdot\lim_{f(x)\to0} p(x)$$ Is squeeze theorem valid for this limit?

$$\lim_{g(x)\to0} f(x)+\lim_{p(x)\to0}f(x)=?$$ (is there any formula for the above limit sum?)

Any other theorem concerning the limit is always welcomed.

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Bijayan Ray
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    $|g(x)|<0$, are you sure ??? –  Mar 15 '19 at 14:34
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    This is an interesting question. Please write the intended formula in the fourth line of your text correctly. – Christian Blatter Mar 15 '19 at 14:36
  • @YvesDaoust I think so because the definition of normal limits goes like this.Being inexperienced if I am wrong please correct me. – Bijayan Ray Mar 15 '19 at 14:44
  • Hem, can an absolute value really be negative ? I am pointing to a typo. –  Mar 15 '19 at 14:49
  • oh sorry got it , is it now good? – Bijayan Ray Mar 15 '19 at 14:53
  • @ChristianBlatter I wrote the possible propositions that came in my mind till now, as mentioned in the question, some other propositions are always welcomed. – Bijayan Ray Mar 15 '19 at 14:55
  • I don't see anything extra gained by it which can't be captured by the existing definition of limit. In what context do you think it would be more useful compared to existing framework of limits? – Paramanand Singh Mar 15 '19 at 14:57
  • I do not doubt the existing definition of limit . Main purpose of the question is that , some text uses this limit directly without giving a slightest remark (for example the riemann integral is often written as limit of riemann sum although the riemann sum is not an explicit function of the mesh or the norm), so my question is whether these limit notation exists or is it my misunderstanding regarding the concept in the textbook? – Bijayan Ray Mar 15 '19 at 15:02
  • Limit of a Riemann sum is a different concept and most books define it separately. – Paramanand Singh Mar 15 '19 at 15:15
  • Recently I came across 'lectures in theory of functions of complex variables' by Fedoryuk and few others (not a very famous book), on page 45 I found the definition of integral of function of complex variable as limit of riemann sum, so that is why I am asking the question . – Bijayan Ray Mar 15 '19 at 17:17
  • @ChristianBlatter Can anyone tell me whether such a limit exists or not ? – Bijayan Ray Mar 27 '19 at 14:52
  • @ParamanandSingh Can anyone tell me whether such a limit exists or not ? – Bijayan Ray Mar 27 '19 at 14:52
  • Well, I have not found such definition in common/ppopular books of calculus / analysis. And as I have said earlier I don't see a need of such definition. – Paramanand Singh Mar 27 '19 at 18:20
  • @ParamanandSingh How is the limit (as said in usual context) of Riemann integral defined ? – Bijayan Ray Mar 27 '19 at 18:22
  • See https://math.stackexchange.com/a/2047959/72031 for the two common definitions. Both of these are completely different from what you propose. – Paramanand Singh Mar 27 '19 at 18:25

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