My favorite involves geometric Brownian motion x. lim N → ∞ lim T → ∞ g: time-average growth rate lim T → ∞ lim N → ∞ g: growth rate of the expectation valuepic.twitter.com/t96ihNvwPz
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My favorite involves geometric Brownian motion x. lim N → ∞ lim T → ∞ g: time-average growth rate lim T → ∞ lim N → ∞ g: growth rate of the expectation valuepic.twitter.com/t96ihNvwPz
1st j/(j+r) = j(1/j+r), so lim j → ∞ lim r → ∞ aⱼᵣ = lim j → ∞ (lim r → ∞ j(1/j+r)) = lim j → ∞ j*1/∞ = lim j → ∞ j*0 = lim j → ∞ 0 = 0 2nd j/(j+r) = 1/(1+r/J), so lim r → ∞ lim j → ∞ aⱼᵣ = lim r → ∞ (lim j → ∞ 1/(1+r/J)) = lim r → ∞ 1/1 = 1 qed
I forgot the brackets in j/(j+r) = j(1/(j+r)) in the first and second line. Sorry.
This one comes up in the analysis of how a metal responds to an electric field!
So, a limit of 1/2 if they are taken simultaneously?
An interesting way to think of it is that taking the limits in different orders promotes different variables as going to infinity "faster", e.g. j>>r or vice versa, in the absence of a more specific relationship, such as a function j(r).
Doesn’t the left assume that j is not infinity?
Can u explain it why? Or point me to the proof...
see above.
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