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

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

    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.
     
  2. jcsd
  3. Dick

    Dick 25,853
    Science Advisor
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    Re: Find a value of constant "k" so a limit to infinity exists

    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.
     
  4. Re: Find a value of constant "k" so a limit to infinity exists

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

    Dick 25,853
    Science Advisor
    Homework Helper

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

    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.
     
  6. Re: Find a value of constant "k" so a limit to infinity exists

    oh my goodness. i just understood it now..thank you *bangs head on desk* :)
     
  7. Re: Find a value of constant "k" so a limit to infinity exists

    so to show that say for k=4 the limit is 0, would I just divided each term top and bottom by x^4?
     
  8. Re: Find a value of constant "k" so a limit to infinity exists

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

    Dick 25,853
    Science Advisor
    Homework Helper

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

    Right on both counts.
     
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