lim (6x+1)/(|6-2x|) as x->infinity is what number? I thought it was 0, but answer key says 3. How?

Answer 1

Get a "feel" for the numbers

look at the |6-2x)
= 2|3-x|
as x ---->∞ , we end up with 2(∞)

at the top , as x ---∞, the +1 becomes less and less signifiant as we approach 6(∞)
so in effect we have 6∞/2∞ = 3
(my notation is really bad and mathematically incorrect, but it gets the idea across)

e.g. let's take x = 1,000,000
we would have
(6,000,001)/1999994 = 3.0000095
of course x = 1,000,000 is not nearly large enough, but as it gets larger our answer would get even closer to 3