So I have:

lim as u-->inf of (-3u^3-6u+6)/sqrt(2-u+6u^2+4u^2)

I know how to work it out, and when I do, I come up with the a result of -inf. The answer key I have says it's +inf.
The reasoning I have is when you take out u^3/u^2, you can simplify to get inf*(-3/2). Shouldn't that make the answer -inf? It's possible my professor messed up and forgot to put the negative sign. Thanks!

sqrt can be positive or negative.

easy way to test it yourself is to pick a few values for u. eg 10, 100, 10000000. and see what you get.

going towards infinity you can just look at the highest powers as a first step to see if that gives you the result
in your case it is -3u^3/srqt(10u^2) which comes down to -u^3/sqrt(u^2)
since it would be standard to use the positive root rather than the negative one, i would have given the same answer you did
but as 'cartblanche' said, the root can be plus or minus - just wondering why your professor picked the negative one
btw, the test proposed by 'cartblanche' does not really help

Completely forgot about the sqrt being positive or negative there lol. Thanks!

The reason I was confused is because this is the key he gave

" sqrt(X) " always denotes the positive root (when it exists) of the equation A² = X.

You were right in the first place.

When using the factorisation (previous post) :

- the first term (u^3 / u²) simplifies as u, and its limit is either +infinity (as u approaches +infinity) or - infinity (as u approaches - infinity).

- the numerator always tends to -3 ( as u approaches + or - infinity)

- the denominator always tends to 2 ( as u approaches + or - infinity).

As a result, the given quantity tends to - infinity as u approaches + infinity, and tends to + infinity as u approaches - infinity.

Notice that the way the problem is asked is unclear ( " u -> infinity " without saying if it is " u -> + infinity " or " u -> - infinity").

Also notice that, whereas the first form exists for any real number u, the factorized form doesn't stand when u = 0 (but there's no problem while considering the limits towards infinity).

if he gave you this picture, he just has been slack to ignore the -3/2 (especially the "-")
so maybe he should go back to school