# Weird result in complex limit

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?

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])

• Thank you, it worked. Apr 28, 2019 at 22:43
• Using Assumptions -> u >= 0 && e > 0 gives the same form as the OP's hand calculation. Apr 28, 2019 at 23:32

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.