At a Glance - In the Real World
Sometimes measurements and results aren't perfect. This is true in science, engineering, and life. How close to 16.9 fluid ounces of Pythagor-ade can a plastic bottle hold in order to allow 16.9 fluid ounces to be printed on the label? There is a certain range allowed by ShmoopCo that is sure to quench calculus induced thirst, but not too much to send a student to the little girls or boys room mid-class. Using limits, we can answer such questions.
One useful aspect of limits is something many calculus classes don't cover, or don't cover much: the error. To understand error, it helps to understand the formal definition of a limit.
Definition. Let f(x) be a function. We say the limit of f(x) as x approaches a is L, written
if for every real number ε > 0 there exists a real number δ > 0 such that
if |x-a| < δ then |f(x)- L| < δ.
We can think of ε as the error that's allowed in a measurement. If we know the measurement is the limit of some "nice'' function, the definition of limit says that we can choose the error ε that we want to allow and there will be some δ that will guarantee our measurement will have only the allowed amount of error.