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Geometry/parallel lines


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?

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


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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