# Functions, Graphs, and Limits

# In the Real World

Sometime measurements and results aren't perfect. This is true in science, engineering and life. How close to 16.9 fluid ounces of Mathlete-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.