What’s the difference between something that approaches infinity and something that is infinite. I’m trying to understand what the difference between something that approaches infinity and something that is infinite is because I was told that that I cant divide infinity by infinity but for some reason I can divide x/x when x approaches infinity 
 A: This is my humble understanding it is not complete, but I hope it is simple enough to give a meaning to an infinitely complex concept.

" I was told that that I cant divide infinity by infinity "

Infinity in Mathematics could refer to a number beyond huge. However, it is not a specific value. There is no starting point to the concept of infinity. Hence the variables $a$ and $b$ may be said to be infinite, but this does not mean they are equal. It is like when you describe to some one two girls and say they are pretty. You don't quantify the prettiness by value (maybe this is a bad example, anyway!).
Since the infinite value is not defined, we can't do algebraic operations on them. You can't add, subtract, multiply or divide and always get a meaningful value consistent with the laws of "Common Algebra". Those laws work on finite numbers and some laws exist for Complex numbers.
Note that you if you apply algebraic operations on infinite quantities, you may get correct answers, however, in this case, it is a coincidence and not the rule. For example when you think of adding two infinite quantities, common sense tells us we get an even larger quantity.
Without adding confusion, Infinity can be used not only to define a limit but also as a value in the Extended Real Number system, this is another Mathematics topic, I would rather leave outside this discussion.

for some reason I can divide x/x when x approaches infinity

According to the discussion above, well, we can't do that. We could attempt to study the behavior of some function like $\frac{x^2}{x}$ as $x$ approaches infinity (that is, as $x$ becomes very large). We say as $x$ grows so very large then $f(x)$ also grows to be very large. 
For short, people say the limit of a function like f(x) above "is infinity" when "x goes to/approaches infinity". However, the real meaning is what we just discussed. The "something" that "approaches infinity" is an expression or a variable we wish to study. It does not have to be huge in itself. For example, what happens to $g(x)=\frac{1}{x}$ as x gets so very large? In fact, when $x$ gets so very large, g(x) gets smaller and smaller and Calculus tells us it becomes exactly zero by using theorems of Limits. The word "behavior" used here is a bit general. As we just showed, sometimes Calculus tells us an exact value not a just a description of behavior. This is also the case when taking the limit of a constant function.
Mathematics study this concept in Calculus and Analysis under the subject of "Limits". 
There many other aspects of "infinity" in Infinite Series and other branches of mathematics. The net is full of discussions on this topic. For example, I hope you find some readable parts in Wiki-Infinity.
A: When you are talking about "something that is approaching infinity" you have in mind a domain $X$, a proper or improper "limiting point" $\xi$ for variable points $x\in X$, and a function $f:\>X\to{\mathbb R}$. Then maybe $f(x)$ will be approaching $\infty$ when $x\to\xi$. 
When you are talking about "something that is infinite" you have in mind a special object $\iota$ that is "infinite" in the considered environment, say $\infty$ in the Riemann sphere $\bar{\mathbb C}$, or one of $\pm\infty$ in $\bar{\mathbb R}$. Such special points are "ordinary points" in some way, but require "exception handling rules" when it comes to limit rules or calculations.
At any rate, "something that is approaching infinity" and "something that is infinite" are completely different things.
A: There is no number that is infinite. When you are working with real numbers, $\infty$ is not one, and you can't use it in any algebraic expression.(1)
The expression $x$ approaches infinity" does not mean $x$ gets closer and closer to some thing called "$\infty$". That's just a shorthand (and sometimes confusing) way to refer to an argument that reasons with values of $x$ that are as large as you please: bigger than $100$, bigger than $1000$, bigger than a google. But you make that argument for each specification of "bigger than". You never talk about $x = \infty$.
(1) The story is a little different when you study the cardinality of sets, but that's not what you are asking about.
A: It's a linguistic difference. Indeed, the word infinity can mean many different things depending on context.
First, you should note that a variable quantity cannot actually be infinite, by definition -- it may only assume absolute values that vary between nothingness and being arbitrarily large. It is the idea of being arbitrarily large in absolute value that we mean when we say a variable becomes infinite, or is infinite. It's just shorthand for the italicised phrase. So, being infinite (or approaching infinity) for a variable real quantity simply means the quantity becomes larger than any preassigned bounds.
From the foregoing it should now become clear that you cannot divide infinity by infinity (in the usual sense) since infinity is not a real number. Indeed such talk does not make sense literally speaking since infinity here is just a short word for the process of exceeding any given quantity. So, you can't divide random properties of variables -- which is what infinity means here. It's just the property of being as large as you please. But again, there is a sense in which we may (informally) talk of dividing infinities. This is just short hand for meaning that we consider the behavior of a quotient of two variables that approach infinity. You can see that it is the actual, finite, quantities at each point of the process that we're dividing, nothing actually about infinity. We only consider the behaviour of this varying quotient as both quantities become arbitrarily large. The behaviour is varied. The quotient may also become infinite, or settle down to a finite amount, or fluctuate. This is why, even when we want to extend arithmetic to include points called infinities, we don't usually define the quotients of infinities.
Thus, it should now be clear that when you divide a nonzero quantity by itself, you're simply doing normal, finite arithmetic. When we do $x/x,$ say, for positive $x,$ the result is $1,$ of course. But now we consider the behavior of this quotient as what you're dividing by itself ($x$) becomes arbitrarily large. It should be clear that the value of the ratio does not change, so that no matter how large $x$ is, we always have $x/x=1.$ We then say that the limiting value of $x/x$ as $x$ approaches infinity (or, informally, when $x$ is infinite) is $1.$
A: The reason $\lim_{x\to\infty}\frac{x}{x}=1$ but $\frac{\infty}{\infty}$ doesn't exist is because the latter could mean $\lim_{x\to\infty}\frac{f(x)}{g(x)}$ for any functions $f,\,g$ satisfying $\lim_{x\to\infty}f(x)=\lim_{x\to\infty}g(x)=\infty$. Then the result could be any $c\in\Bbb R^+$ you like with $f=cx,\,g=x$, or $\infty$ if $f=x^2,\,g=x$, or $0$ if you reverse that, or it could be undefined if e.g. $f=x(2+\sin x),\,g=x$.
A: Concerning $∞$, you can read Christian Blatter's answer or Ethan Bolker's answer. However, the main issue with your inquiry can be expressed completely without reference to $∞$.

"$0/0$" (note the quotes!) is a meaningless expression, even though $x/x = 1$ for every positive real variable $x$, even if $x$ is decreasing to $0$ (in some limiting situation).

The point is that when you have a limiting situation, you have a variable that is eventually staying arbitrarily close to but never equal to some limit point. When we say "as $x → 0$", the variable $x$ tends to but is never zero, so there is absolutely no problem with "$x/x$" and we have $x/x = 1$.
