Recently, a question that I posted raised a healthy debate.
The question was to find .
There were various veiwpoints, none of them illogical.
Viewpoint 1 : It is complete gobblydook
The given problem can be written as :
However, exchanging the limits,
depending on or
Hence, even if we consider only the left OR right limit, there is ambiguity.
Hence, it is gobblydook.
Viewpoint 2 : The Left is not Right
No, No... You are not to make a mess of the problem by taking two 'limits', it is the same quantity that is varying.
The right limit is (Courtesy : Ryan Tamburrino)
Applying L'Hopital's rule,
The left limit involves complex exponentiation, fractional power of negative numbers and all other mathematical boggarts. It would be a bad idea to go into that lest we fall prey to tetration.
Viewpoint 3 : All's well - The limits agree
The right limit is indeed 1, as shown in Viewpoint 2. However, there is no need to panic as the left limit is also a tame one, and certainly not a boggart.
For the left limit, consider with being positive. The problem now becomes
Whom do you agree with?
What is your opinion on the 'value' of . (Not many seem to want to fix a value for it!!).
What is your opinion on the existence/agreement/value of the limits?