Question

What is the precise definition of a limit?

Answer 1

There are multiple precise definitions of a limit depending on the context. If you're talking about a standard introduction to calculus, though, the formal definition you're looking for goes something like this:
A function \(f\) approaches a limit \(l\) near \(a\) means that, for every \(\epsilon > 0\), there is a \(\delta>0\) such that, for all \(x\), \(0<|x-a|<\delta\implies|f(x)-l|<\epsilon.\)

Answer 2

If f(x) gets closer and closer to a number L as x gets closer &closer to c from either side , then L is the limit of f(x) as x approaches c.The behaviour is expressed by writing
\[\lim_{x \rightarrow c}f(x)=L\]