I'm starting to think you can always apply them, but you might get indeterminate form 0/0. However, I haven't seen this stated in relation to limits, only in some complex derivatives. It's not stated when limit properties are applied.
I had expected that you could apply properties and rules blindly (with any preconditions included as part of the property/rule), but maybe you must also look at the outcome you get in order to know what rules you're allowed to apply? Perhaps it's like division in algebra, where a manipulation might be valid (at "compile time"), but if specific values of variables make a divisor equal zero, then it's undefined (at "runtime")?
I'll give a warmup example, then the one I'm actually concerned about.
1) First example: applying the quotient limit property to x/x as x→0, which should be 1:
limx→0xx=limx→0xlimx→0x=00
2) Second example: in part of a proof of the product rule (used by Khan at 7:30, and Spivak in Ch.10, Theorem 4, p.45):
limh→0f(x+h)[g(x+h)−g(x)]h=limh→0f(x+h)limh→0g(x+h)−g(x)h
Then using limh→0f(x+h)=f(x):
=f(x)limh→0g(x+h)−g(x)h
My difficulty is you could apply the same approach to the other factor: limh→0g(x+h)=g(x).
limh→0f(x+h)hlimh→0[g(x+h)−g(x)]=limh→0f(x+h)h[limh→0g(x+h)−limh→0g(x)]=limh→0f(x+h)h[g(x)−g(x)]=limh→0f(x+h)h0=0
Am I wrong to think you should be allowed to apply property limits whenever you like, without condition? (And if there are pre-conditions, they should be part of the property?) Or, is there some aspect relating to 0/0 (indeterminate form) whose connection to property limits I've somehow missed or not appreciated?
It seems derivatives are about finding a ratio a/b, even as a,b→0. But I think limits are meant to be standalone, independent of derivatives.
PS. if my confusion is too fundamental to address in an answer, could you suggest a reference that definitely does address it, please? (It's a lot of work to go through a reference, only to find it doesn't have the answer I need.) Many thanks!
No comments:
Post a Comment