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Topic: [ap-calculus] e
Replies: 1   Last Post: Oct 23, 2012 5:44 PM

 Jon Stark Posts: 914 Registered: 6/10/06
RE: [ap-calculus] e
Posted: Oct 23, 2012 5:44 PM

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Your calculator simply doesn't keep enough digits.
Pick a big x value, such as 10 ^ 15. The exponent will be represented properly in the calculator something like 1.000000000000 E 15, keeping 12 (or perhaps 14, depending on model) significant digits in a form equivalent to scientific notation.
But the part of your expression in parentheses is (1 + 10 ^ (- 15)) which must be computed by addition.
To get that value, the calculator tries to add 1.00000000000 to .000000000000001 (represented internally as 1.00000000000 E 0 and 1.00000000000 E -15), which SHOULD be 1.000000000000001. Note that the final "1" is very, very far from the leading "1" in that sum. Since the calculator can't keep more than 12 digits, it truncates that down to 1.00000000000 and your expression is really evaluated as just (1) ^ (10^15) instead of the computation you meant to do, so the result will obviously and erroneously approach 1.
If your calculator could keep an enormous number of digits, you could get answers approaching e with larger x values, but eventually you would reach a point at which the sum would truncate as above.

So I had a pre calculus student ask me why the lim x->inf (1+1/x)^x is e. I can prove it, but I've never noticed until he showed me this that in the calculator, the graph of (1+1/x)^x actually goes to one when you plug EXTREMELY large numbers into the table values. Does anyone know how to explain this?

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