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  • 3.4 Another perspective on indeterminate forms

    An expression like 0/0, called an indeterminate form, can be thought of in a different way in terms of infinitesimals. Suppose I tell you I have two infinitesimal numbers d and e in my pocket, and I ask you whether d/e is finite, infinite, or infinitesimal. You can't tell, because d and e might not be infinitesimals of the same order of magnitude. For instance, if e=37d, then d/e=1/37 is finite; but if e=d2, then d/e is infinite; and if d=e2, then d/e is infinitesimal. Acting this out with numbers that are small but not infinitesimal,

    eq_c2eda4fd.png

    eq_6160a5a2.png

    eq_be270f80.png

    On the other hand, suppose I tell you I have an infinitesimal number d and a finite number x, and I ask you to speculate about d/x. You know for sure that it's going to be infinitesimal. Likewise, you can be sure that x/d is infinite. These aren't indeterminate forms.

    We can do something similar with infinite numbers. If H and K are both infinite, then H-K is indeterminate. It could be infinite, for example, if H was positive infinite and K=H/2. On the other hand, it could be finite if H=K+1. Acting this out with big but finite numbers,

    eq_6cb725cb.png

    eq_ca3744b3.png

    Example 14

    ◊ If H is a positive infinite number, is √H+1-√H-1 finite, infinite, infinitesimal, or indeterminate?

    ◊ Trying it with a finite, big number, we have

    eq_2f9e3026.png

    eq_05a8af27.png

    which is clearly a wannabe infinitesimal. We can verify the result using Inf:

       : H=1/d
      d^-1 
       : sqrt(H+1)-sqrt(H-1)
      d^1/2+0.125d^5/2+... 
    

    For convenience, the first line of input defines an infinite number H in terms of the calculator's built-in infinitesimal d. The result has only positive powers of d, so it's clearly infinitesimal.

    More rigorously, we can rewrite the expression as √H(√1+1/H-√1-1/H). Since the derivative of the square root function √x evaluated at x=1 is 1/2, we can approximate this as

    eq_f1315969.png

    eq_3d668305.png

    eq_0191fd22.png

    which is infinitesimal.