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Functions, Graphs, and Limits

Functions, Graphs, and Limits

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.

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