How can I do this?

CPhill, one of the moderators might be able to help you in this question, but is not online right now.

As x → inf x⁴ dominates, so tan^-1(x²-x⁴) →  tan^-1(-x⁴)  

Now tan is opposite over adjacent and this goes to infinity as adjacent goes to zero.  As the adjacent goes to zero the angle goes closer and closer to pi/2 (90°).  This means in the limit as the argument goes to infinity, the angle whose tangent is infinity is pi/2.  In the case we have here, the argument goes to -infinity, so the angle is whose tangent is -infinity is -pi/2.