Limits and Errors

Prologue to Section 1:

Imagine this circle:

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This is a circle with a radius of three and a center at (0,0). And finding its area is simple as well, if you use the famous formula to find the area of a circle:

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Throughout this introduction to limits, constantly keep in mind that the area of the circle to four decimal places is 28.2743.

Section 1:

For those of us who have taken basic geometry classes know what it means to inscribe a polygon into a circle, so we will begin by inscribing a simple triangle into our original circle.

With the careful calculations shown below, it is possible to easily get the area of the triangle in order to estimate the area of the circle:

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in which n is the number of sides and r is the radius

We do this in order to approximate the area of our original circle, although it may be a poor estimation. The next few steps show what happens when we increase the number of sides of the inscribed polygon, and how our estimation nears 28.2743 or 9π. Eventually, the graphs of the regular polygons are too close to the exact circle to graph, so we will just have you imagine the drawings.

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Imagine an 80-sided figure:

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Now imagine a 1000-sided figure:

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And finally a one million sided figure:

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As you can tell, when we have a one million sided figure, the estimation was exact to the four decimal places, far more accurate than the original triangle.

Section 1 Conclusion:

Let’s refer to the number of sides on this n-gon to be n. Here are some true statements about this n-gon:

· As n gets larger, the n-gon gets closer to being the circle.

· As n approaches infinity, the area of the n-gon approaches the area of the circle.

· The limit of the n-gon, as n approaches infinity, is the circle

This can be shown by a simple notation:

The n-gon will never theoretically get to the exact area of the circle, but it will get as close as you want it to! It actually gets so close that it might as well just be the exact area of the circle, that’s the theory behind limits.

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Archimedes used this method to find the area of circles before the value for pi was ever discovered, how cool is that?



Section 2:

Here’s another example, this time using a numerical approach rather than the geometrical approach we have just explained.

Imagine a sequence that is written as:Limits14

When will this sequence ever reach one?

The table will look somewhat like this:

Value of x

Value of f(x)

1

0.5

2

0.67

3

0.75

4

0.8

As you can tell, f(x) gets closer and closer to 1. If you recall from section one, the limit theory states that eventually, f(x) will reach 1. This statement can be expressed as:

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In English, the above statement is read aloud as: The limit of x/(x+1) as x goes to infinity equals 1.

Section 3:

If the table in section 3 wasn't sufficient in order to explain the concept of limits, we will present a graph of :

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In the graph above, the visible graph never reaches the f(x)=1 mark, represented with the vertical red line. But as x approaches infinity, the graph will, at the infinity, reach f(x)=1 This line can be referred to as an asymptote, or as we have earlier referred to it, as a limit.

Section 4:

At first, it is difficult to see how it can eventually reach its limit, but this is the concept of infinity. Here is a more simple mathematical way to show that it actually does reach the limit of 1:

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Limits Conclusion:

As we have shown throughout this introduction to limits, limits are a broad and undeniably an interesting subject to study. But as for now, we will leave you wondering more about limits, so when you study them more in depth you will have a head start!