A few problems with the evaluation of limits of multivariable functions

Questions
,
1

Recently, I have been studying limits of multivariable functions.

In particular, I am interested to evaluate the expression

\displaystyle \frac{1}{(4 \left((\alpha + 2 y\beta + x \delta)^2 + (\epsilon +2 y \eta + x \lambda)^2\right)^2 \left(\left(y \delta + a_{10} + 2 x a_{20})^2 + (y \lambda + b_{10}+ 2xb_{20})^2\right))^2\right)}.

in the neighborhood of y=0 and

x=\displaystyle \frac{(-\delta \epsilon \sqrt{-(\delta \eta - \beta\lambda)^2} \ + \alpha \lambda \sqrt{-(\delta \eta - \beta \lambda)^2} - \ \delta \eta (\alpha \delta + \epsilon \lambda) + \beta \ \lambda (\alpha \delta + \epsilon \lambda))}{((\delta \ \eta - \beta \lambda) (\delta^2 + \lambda^2))}.

By employing the Mathematica, one retrives \textit{ComplexInfinity} as a result. Based on the above,

  1. What does \textit{ComplexInfinity} mean here? That is to say, what is the suppose direction of a such result (e.g. -\infty or +\infty or just \infty ?
  2. Is there another way to evaluate that limit and obtain a more reliable result through Mathematica or Maple?

Ps: I have read other posts on the subject, however, these did not clarify my doubts.

2

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3

Your function is strange. It contains the square root of minus a square. Are your variables complex? If so, you should not be surprised to get ComplexInfinity as output. Also, you have a lot of parameters, which often make symbolic calculation very complicated.

4

Assuming the denominator vanishes for that {x,y} pair, probably Indeterminate would be more correct. But what Gianluca Gorni noted is on target, limiting values will depend on patch of approach.

5

Thanks Gianluca Gorni for commeting! In principle, all coefficients are real. Nevertheless, I will revise the expressions according to your and Mr. Lichtblau’s comments.

6

Thanks for commenting! On account of your observations, I will investigate these expressions further.