Limits - Differential Calculus

Question

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?

Answer 1

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)

Answer 2

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.