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What I really want to ask is simple, and something which deals with the concept of limits and not really problem solving. So my question is

Do limits exist only for functions? Or do they exist for other mathematical expressions like fractions, or some special types of series?

For example, if we have the Fibonacci sequence of numbers $(1,1,2,3,5,8,\dotsc)$, can we define a limit as the following $$ \lim_{n\to\infty} \frac{A_n}{A_{n-1}} = \text{Golden Ratio}= 1.618033\dots $$ because here we don't have any function and still I am applying a limit to it. So is that possible for mathematical expressions other than functions?

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  • $\begingroup$ @AhmedS.Attaalla Doesn't help to notice that the second term has negative base, so function is undefined over the reals not including the naturals. $\endgroup$ – Simply Beautiful Art Jan 1 '17 at 18:38
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    $\begingroup$ Sequences are functions. In fact the nth term of the Fibonacci sequence has closed form $f_n=\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\right)^n- \frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^n$. $\endgroup$ – Ahmed S. Attaalla Jan 1 '17 at 18:38
  • $\begingroup$ @AhmedS.Attaalla But then your comment isn't very obvious. Whether or not your $f_n$ is $\mathbb N\to\mathbb N$ is not immediately clear, nor is it very clear the limit of ratios be $\phi$. $\endgroup$ – Simply Beautiful Art Jan 1 '17 at 18:44
  • $\begingroup$ Counterexample: Limits do not exist for you because you are Limit-less. Badabim! $\endgroup$ – Billy Rubina Jan 5 '17 at 9:10
  • $\begingroup$ Now, not talking shit: I guess you're confusing something: What is the meaning of "having a function"? Having a function for the Fibonacci numbers in terms of recurrence is "having a function", the rule is just a little bit different of having - for example - $f(x)=\log x$. $\endgroup$ – Billy Rubina Jan 5 '17 at 9:13
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In general, limits are defined on metric spaces. More generally, limits are defined in topological spaces (every metric space is a topological space, but the converse is not true).

A metric space is a set $X$ along with a distance function $d$ that assigns a distance (a nonnegative real number) $d(x,y)$ between any two points $x,y$. A limit tries to captures the idea of a sequence of points $(x_n)_n$ approaching a particular point $x$, written $x_n \rightarrow x$. The notion of "approaching $x$" is made rigorous through the distance function.

Limits of functions can be defined using sequences: We say that $f(x^\prime)\rightarrow y$ as $x^\prime \rightarrow x$ if given any sequence $(x_n)_n$ such that $x_n \rightarrow x$, $f(x_n)\rightarrow y$. You may have seen a different (but equivalent) definition in your classes/books.

(I would learn about metric spaces before moving onto topological spaces)

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  • $\begingroup$ however may limits don't exist, unless cauchyness of the sequence and completeness of the metric space is granted $\endgroup$ – janmarqz Jan 1 '17 at 19:42
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They exist in sequences, like the one you have shown (one of my favorite actually).

Limits can always be applied to any scenario that has the epsilon-delta definition of a limit applicable. e.g.

$$\left|\frac{a_n}{a_{n-1}}-\phi\right|<\epsilon$$

where $a_n$ is the $n$th Fibonacci number. I leave it to readers to attempt to finish this limit verification.

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