Overview: The idea of limits underlies almost all we do in
calculus

In most previous math classes, we have learned how to get exact
answers. If we want to solve $x^2 - 5x + 6 =0$, the answer
isn't "close to 1.99" or "close to 3.01". The quadratic
formula tells us: "$x$ is exactly 2 or exactly 3".

In calculus, we have problems where we can't get an exact answer
directly. Instead, we find an approximate
answer, then a better answer, then an even better answer.
The exact answer is the limit
of these approximations.

A statement of a limit is "the limit as $x$ approaches (some $x$
value) of the function $f(x)$ is exactly equal to (some $y$ value),
which we write as $$\lim_{x \to \tiny\hbox{(some $x$ value)}}
f(x) = \hbox{(some $y$ value)}.$$ For example,
$$\lim_{x\to 5}(x^2-2)=23.$$
This is the most important idea in all of calculus. You will need to
learn it well as you work through understanding limits.