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 x2−5x+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 limx→(some x value)f(x)=(some y value).
For example,
limx→5(x2−2)=23.
This is the most important idea in all of calculus. You will need to
learn it well as you work through understanding limits.