Inspired by Anandhu Raj
( n → 0 + lim x → 0 + lim x n ) − ( x → 0 + lim n → 0 + lim x n ) = ?
Inspiration, see Anandhu Raj's comment in Ivan's solution
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2 solutions
[This is not a solution]
The reason why the answer is not 0, is because we cannot simply interchange the order of limits, even though these functions are continuous. That is because, the pointwise limit of the functions is not continuous, resulting in this difference. In particular, n → 0 + lim x n = { 1 0 0 < x ≤ 1 x = 0
This begins to delve into the area of analysis (much more rigourous calculus), which deals with continuity and convergence.
Well, you did not really simply change the order of the limits . You added a + too!
By the way, the inspiration problem does not redirect to Anandhu Raj's problem
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Thanks, they were all supposed to have a +, to avoid dealing with negative/complex exponents. (fixed)
The inspiration was from a comment in a solution. Sometimes, the link to a solution comment doesn't work well, because you first land on the problem. I have made this more explicit. Thanks!
the first limit = 0 because as x → 0, x n = 0 regardless of n , but the second limit = 1 because as n → 0, x n = 1 regardless of x as both never actually reach zero. [This is not intended to be the most rigorous solution]