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Welcome, Shmooperinos, to the final section in the final chapter of Precalculus. Insert round of applause here. This last one is all about where we can find and use limits outside of our pesky math textbooks. And trust us, there are far too many applications to even scratch the surface here.

But who says we can't try?

From the growth and decay of bacteria to the population of squirrels in your neighbor's backyard, functions can model all sorts of things. But what about when you want to know what happens a long time from now? How many squirrels can that giant oak tree really hold before the squirrels start looking elsewhere? How do we get there?

While we don't have all the answers, we know limits can help.

First, we need to know or be refreshed about one our favorite equations: the equation for logistic growth.

We like it for what it can do, not necessarily how it looks. All those letters stand for:

*y *= the population of whatever we might be analyzing

*L* = the carrying capacity

*k* = a constant

*t* = time

Okay, we admit that's a lot to take in and a little bit confusing to boot. Let's give you some context to help things along.

Suppose we had a population of Emperor Penguins in Antarctica, which we learn, after careful study, can be modeled by the equation:

The question here isn't so much how many penguins are hanging around right now or even how many there will be in 10 years. After all, that just becomes a simple substitution problem. The real question we'd like to answer is will the penguin population grow without bound or level off? Based on our model, how many penguins can Antarctica support?

This sounds an awful lot like end behavior to us. So lets take the limit as* t* approaches infinity.

This is kind of a tricky limit to evaluate. Lucky for us, we learned some sweet properties of limits that allow us to think about this thing as:

Since *e* raised to a giant negative number definitely approaches zero, we know this limit approaches 400,000 Emperor Penguins. Good thing too, because the carrying capacity is actually defined as the maximum population that the model can sustain. Phew, that was close.

Limits can do other things besides just talk about functions. What about if we looked at polygons? Yes, polygons. More specifically, lets look at regular polygons with a radius of 1 unit. Here are two of them:

Believe it or not, all regular polygons can be said to have a radius. And we could continue to draw these shapes until we were blue in the face. Or fingertips. Triangle, square, pentagon, hexagon, heptagon, octagon; the list goes on and on and on.

The stranger thing, however, occurs when we start to look at their areas and perimeters. The table below uses *n* as the number of sides and then has the area and perimeter for each regular polygon rounded to two decimal places.

n: | 3 | 4 | 5 | 8 | 10 | 20 | 50 | |

Area: | 1.30 | 2 | 2.38 | 2.39 | 3.09 | 3.13 | 3.14 | |

Perimeter: | 5.20 | 5.66 | 5.88 | 6.12 | 6.18 | 6.26 | 6.28 |

Now this is the mind-bendy part. What happens to the general shape of a polygon as we add more sides? It slowly but surely starts look more and more like a circle.

And check this out, what does the area of a polygon approach as the sides increase to infinity? π. Exactly. Also known as π*r*^{2} which in this case is π(1)^{2} = π.

And the perimeter, or should we say circumference? 2π*r*. For here, that's 2π(1) = 2π.

Holy smokes. This is really happening right now.

Okay, okay. We admit this isn't the most real world thing in the entire universe. But it begs the question: is a circle really a polygon with an infinite number of sides?

#mindblown