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Geometry/parallel lines

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Question
Hi Scott,

I was taught in school,...a long time ago that two parallel lines meet at infinity. Can that be proved mathematically? How does that gel with the geometric
infinite series 1/1-x which only converges at infinity?
Thanks

Answer
By math, two parallel lines never intersect.  Look at y=0 and y=1.  They are parallel,
but as x increases, the relative difference of 1 in y appears to get smaller
when compared to the size of x.  

When it is said that 1/(1-x) converges to 0 as x goes to infinity means that no matter how small the difference between the number and zero is wanted, an x big enough can be found to make it that small.

For example, since it is known to converge to 0, given any number, no matter how small,
a number n can be found to make the absolute value smaller than the size desired.

If it is to be smaller than 0.1, any n>=11 would work.
If it is to be smaller that 0.01, any n>=101 would work.
If it is to be smaller than 0.001, any n>=1001 would work.

For a limit to exist, no matter how small a choice is chosen,
there exists a value for n such that if x is bigger than that, it works.

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Scott A Wilson

Expertise

I can answer whatever questions you ask except how to trisect an angle. The ones I can answer include constructing parallel lines, dividing a line into n sections, bisecting an angle, splitting an angle in half, and almost anything else that is done in geometry.

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I have been assisting people in Geometry since the 80's.

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I have an MS at Oregon State and a BS at Oregon State, both with honors.

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I was the outstanding student in high school in the area of geometry and math in general.

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Over 8,500 people, mostly in math, with almost 450 in geometry.

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