crazy limit!!

I haven't seen anyone explain why this is wrong yet' date=' even though everyone seems to agree that the limit is 2.

 

What you're doing is factoring out an x from the square root, so you get this:

 

[math'] \lim_{x\to -\infty} [x + |x| \sqrt{1+\frac{4}{x}+\frac{1}{x^2}}] [/math]

 

So far so good. But then you're saying that what's left under the square root sign is going towards one as x goes to negative infinity, so that the limit above should go towards [math] \lim_{x\to -\infty} [x - x] = 0 [/math]. This would be true if x were finite -- but since x in this case is approaching infinity, multiplying it with something that's just approaching one doesn't necessarily leave it unchanged. It could converge to anything, or diverge, for all we know.

What are you saying?