Find a value of constant "k" so a limit to infinity exists

madgab89

1. The problem statement, all variables and given/known data

Find a value of the constant k such that the limit exists.

2. Relevant equations

lim x to infinity of

x^3-6/x^k+3

3. The attempt at a solution
I started by setting the equation equal to infinity and attempted to rearrange it but got pretty much nowhere. I also broke it up using limit rules and also ended up nowhere.

Dick

Can you guess what a k might be? For the limit to exist the numerator and denominator have to grow at similar rates as x->infinity.

madgab89

So would k be 3? I'm not sure I'm understanding..

Dick

Yes, k=3 is one value that works. What's the limit in that case? Can you show if k>3 the limit is 0 (so it exists) and if k<3 the limit is infinity (so it doesn't exist)? You were just asked to find 'a value'. There are lots of choices.

madgab89

oh my goodness. i just understood it now..thank you *bangs head on desk* :)

madgab89

so to show that say for k=4 the limit is 0, would I just divided each term top and bottom by x^4?

madgab89

and for k=3 the value of the limit would be 1, correct?

Dick

Right on both counts.