What is a limit really?

In many fields, especially ones related to the physical sciences, small angle approximations are often used. For example, when x is very small, the sine of x is said to equal x and to equal the tangent of x. These conjectures are then used as theorems in a proof.  In my opinion, this is bad calculus. Such approximations generally serve their purpose well, but are merely a way to make a very long proof look very short.  In calculations, it’s fine to be a little bit off, but not in proofs.  We often come upon a difficult matter to prove and a professor takes the shortcut. While these shortcuts do work, they bring us to the right answer, they are not the real reason why that answer is right. In reality, the sine of x is never equal to x or the tangent of x unless x is absolutely zero.  Consider the following proof:


1. Because the recursion can approach infinity at any rate (even an infinite rate), and in application, x is assumed to be a finite constant just greater than one, this indeterminate quantity (0^inf) evaluates to infinity.  Thus, as we should expect, the anti-proof does not apply to an infinitely small x value or a zero x value.

2. At this point, one may evaluate the expression on the left by flipping and multiplying by the bottom, thus getting sin(0) = 0, which is true, but there is nothing stopping us from moving the cos(x) to the right side of the equation before the self-referencing recursion and thus ending with it on the other side, so the move of multiplying both sides by 1/0 is legal in this case, this way of doing it just makes it more clear.  Of course, if one did start with the cosine on the other side, the proof might also evaluate to 0 = inf.  Admittedly, this proof is somewhat ambiguous over all for these reasons, but the end result (ignoring all the paradoxes) is that a limit is a limit, and any finitely small angle will not satisfy this equation just as much as zero does not equal one, or infinity does not equal zero.  If you plugin an infinitely small value for x, all the problems immediately dissolve and the limit is proven.

3. because, as was said in the previous note, the infinite quantity can approach infinity at any rate (an indeterminate rate), here it could be said to be approaching it at the same rate that x is “approaching” zero (even though x is a finite constant), and thus the limit holds (if we want it too…ha, ha, ha).  Also note, it doesn’t present a problem for us that the power of the cosine had to approach infinity at a rate that would make the power of the cosine evaluate to an infinite quantity, because by continuing to manipulate that original rate of approach, we can also manipulate the rate at which the power of the cosine approaches infinity (which is what we need in this last step).

My Cartesian Point?

My point ultimately with all this silliness that no one actually knows (though we do, I think, know more about infinity and zero and the imagination of numbers– ok, i need to get better jokes–than we say we do) is that if you zoom in far enough (as we did here using the infinite recursion), the tangent of x, the sin of x, and x are just as far away from each other when x doesn’t equal zero as zero is from infinity.  Thus, while the approximations are useful when dealing with actual quantities, they are just bad math when used in proofs (even though everyone does it… it’s pretty much like jumping off a cliff of infinite height), because in proofs, we depend on expressions being absolutely equal, and finitely small number is infinitely larger than an infinitely small number.

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