Question

How can I evaluate this limit?

Answer 1

The answer is the limit approcaches negative infinity. It should be obvious based on the fact that the negative side has (x^4 +2)^3/2 compared to the x^4 on the positive side.
One way to show it is by expanding (x^4 + 2)^3/2 to
(x^4 + 2)^2 • (x^4 +2)^1/2 , and then since (x^4 + 2)^1/2 is always positive and since it multiplies with the minus part,
We can show that (3/4 x^4 )-1/2(x^4 +2)^2 is bigger than the original question without the +2015 part. now we can evaluate the limit and we will get that it diverges to negative infinity, and therefore the original question also diverges to negative infinity since its always smaller than the limit we evaluated(the+2015 part doesn't make a difference).

Answer 2

When x is unbounded, the +2 is no longer significant, so eliminate that and see what you get

Answer 3

~~If it's infinity - infinity, that's 0~~, that's fine. It might not be though, which one of those things is approaching infinity faster, term 1 or term 2?

Edit: poorly explained, that's not untrue, but only sometimes, read the second part first before considering that.

Edit: I'm just past caring but for what its worth, this is actually the exact same thing these other right answers are saying, all you downvoters lol...