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>>> Wouldn't that be increadibly unstable, numerically?
>>
>> How could it be? If you're taking the limit of something that's going
>> to a finite value in a continuous curve, how could your calculation of
>> the value closer to the limit have more error than the error
>> calculated farther from the limit?
>
> According to Google calculator, sin(1e-14)/1e-14 = 1, but
> sin(1e-15)/1e-15 = 0.
Indeed, this was going to be my example. When x is small, sin(x) is
approximately equal to x. But whatever, sin(x)/x when x is small
involves division by a tiny number - which is equivilant to
multiplication by a huge number. It's numerically unstable.
(Of course, the limit of sin(x)/x as x approaches zero is just 1, which
you don't need to "estimate" in the first place. But if you had a
similar formula which approximates some irrational quantity...)
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