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Let's start with a simple example.
Suppose you have a function f(x) = x/x.
You'd be tempted to simplify it to f(x)=1 but that does not take into account when x=0.
This has lead some mathematicians to a small existential crisis because you cannot say that f(0) = 1.
That is until some mathematicians developed and formalized the notion of limits, which is a way of saying that when x approaches some value, then f(x) would approach some other value.
In our example, even if x gets extremely close to 0, f(x) would still be 1:
Limits could be seen as a way to transcend (aka "cheat") the limitations(pun not intended) of "classical" mathematics.
The tricky thing is that the limit must be evaluated 2 times, the first time when x is a little bit smaller than the desired value and the second time when x is a little bit larger.
Why is that? f(x) equals 1 no matter if x is a bit smaller or larger than 0.
Let's take an another example: g(x)=1/x.
We want to see what happens when x approaches 0.
When x is a little bit smaller that 0, g(x) approaches to the negative infinity.
When x is a little bit bigger that 0, g(x) approaches to the positive infinity. The formal equations can be written as follows:
Since the value of g(x) around 0 depends from where x is approaching, we conclude that g(x) has no definite limit around 0.
Limits are also used for when we want to know what happens when x tends to infinity. For f(x) it is still equal to 1 while g(x) approaches 0:
Here you don't need to see whether x is a little smaller or bigger than infinity since infinity is a concept rather than a specific number.
That's all you need to know really
Hope that helps
