Limits help
I need to find this limit:
where a>b>0. I got log(a)-log(b) using the power series representations of a^x (rewrote as exp(x*log(a)) ) and b^x (likewise with b^x). I then subtracting them, divided by x and found the answer. But I am wondering are there alternative ways of doing it e.g. substitution/sandwich (or pinching) method? I think it would be good to know if there are other methods as I want to see if I am missing a more elegant/cleaner solution.
NB: I actually haven't learnt the l'hopital rule in class yet so I that method is ruled out (even though of course it is still valid).
where a>b>0. I got log(a)-log(b) using the power series representations of a^x (rewrote as exp(x*log(a)) ) and b^x (likewise with b^x). I then subtracting them, divided by x and found the answer. But I am wondering are there alternative ways of doing it e.g. substitution/sandwich (or pinching) method? I think it would be good to know if there are other methods as I want to see if I am missing a more elegant/cleaner solution.
NB: I actually haven't learnt the l'hopital rule in class yet so I that method is ruled out (even though of course it is still valid).
You don't need power series or L'Hôpital's rule for this (in fact, it wouldn't be valid to use L'Hôpital since the denominator tends to 0 but the numerator doesn't). In fact, the numerator is bounded, which should make the limit quite obvious. [Or did you mean the limit as ? In which case, consider the dominant term.]
Original post by nuodai
You don't need power series or L'Hôpital's rule for this (in fact, it wouldn't be valid to use L'Hôpital since the denominator tends to 0 but the numerator doesn't). In fact, the numerator is bounded, which should make the limit quite obvious. [Or did you mean the limit as ? In which case, consider the dominant term.]
wrong - numerator tends to zero, since anything to the power of 0 =1 and thus 1-1=0
the limit is Ln[a/b]
(edited 12 years ago)
Original post by nuodai
You don't need power series or L'Hôpital's rule for this (in fact, it wouldn't be valid to use L'Hôpital since the denominator tends to 0 but the numerator doesn't). In fact, the numerator is bounded, which should make the limit quite obvious. [Or did you mean the limit as ? In which case, consider the dominant term.]
But surely the numerator tends to 0 as does the denominator.
Original post by Hasufel
but - how CAN you get the limit? - other than L`Hops rule? - indefinite integral?
Well if you have an expansion of a^x in the form of e^(xlna), then you will find that the x on the denominator will cancel out creating a leading term
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