Weird result in complex limit

Question

I am trying to evaluate a limit:

gamma[w_] = Sqrt[-(u*e)w^2 + I*(u*s)w];
Limit[Re[gamma[x]], {x -> DirectedInfinity[1]}]

I calculated the limit by hand, and the correct answer is (I also checked numerically for some examples of $\{u,e,s\}$ using the software):

$\qquad \frac s2 \sqrt{\frac ue}$

But for some reason, when using Limit, I get

{DirectedInfinity[(Sign[e]^2 Sign[u]^2)^(1/4)]}

So my questions are:

What is going here?
What issues should I be aware of when using Limit?

Answer 1

I think it's worth reporting the issue to support. If you give appropriate assumptions, then you get your expected result:

Limit[Re[gamma[x]], {x -> Infinity}, Assumptions -> u>0 && e>0]

(s u)/(2 Sqrt[e u])

Answer 2

The biggest difference between your hand calculation and the computation performed by Mathematica is that your hand calculation assumes $u$ and $e$ are nonnegative reals. Examples of how this produces different results:

• $u = 1$ and $e = -1$: The limit of Re[gamma[x]] is $\infty$, but your formula gives an imaginary number. A similar thing happens with $u = -1$ and $e = 1$.
• $u = 1$ and $e = 0$: The limit of Re[gamma[x]] is a directed infinity, directed along $\mathrm{Re} \sqrt{\mathrm{i}s}$, which could be $-\infty$, $0$, or $\infty$, depending on the complex argument of $s$. (Mathematica misses this case in the answer you are seeing. Some insight comes from looking at the leading order term in ComplexExpand[Re[Sqrt[-(u*e)w^2 + I*(u*s)s]]], which is $(e^2 u^2 w^4)^{1/4}$. Of course, when $e = 0$, this term is suppressed and then the leading term is $(s^2 u^2 w^2)^{1/4}$. Note that ComplexExpand assumes variables are real unless it is explicitly told otherwise, so it assumes more than we have explicitly established.)
• $u = e = -1$: The limit of Re[gamma[x]] is $-s/2$, but your formula gives $s/2$.
• $u = e = \mathrm{i}$: The limit of Re[gamma[x]] is $\infty$, but your formula gives $s/2$.
• $u = \mathrm{i}, e = 0, s = 1$: The limit of Re[gamma[x]] is $0$, but your formula involves division by zero.