1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

This was on my test

  1. Mar 16, 2006 #1
    Hey everyone,
    I had a math test yesterday. It was pretty hard... This is one of the limits I wasn't able to do.

    [tex]\lim_{x \rightarrow k} \frac {x\sqrt{x} - k\sqrt{k}}{x^{4}-k^{4}}[/tex]

    I tried the Hopital rule, I tried multiplying the whole expression with the denominator. I didn't get to anything better.

    Anyone knows how to do this kind of limits ? Thank you!
     
  2. jcsd
  3. Mar 16, 2006 #2
    Hint: Divide numerator and denominator by x-k.
    Then use
    [tex]\lim_{x \rightarrow k} \frac {x^n-k^n}{x-k}=nk^{n-1}[/tex]
    for both num and den.
     
  4. Mar 16, 2006 #3

    Curious3141

    User Avatar
    Homework Helper

    Factorise the denominator. (x^4 - k^4) = (x-k)(x+k)(x^2+k^2). The rightmost two factors can be evaluated immediately at the limit, they become (2k) and (2k^2) respectively, yes ?

    Then the limit becomes

    [tex]\frac{1}{(2k)(2k^2)}\lim_{x \rightarrow k} \frac {x^{1.5} - k^{1.5}}{x-k}[/tex]

    Now observe that the limit that's left is of the form 0/0, and can be reduced by LH rule. Just differentiate numerator and denominator wrt x. Put x = k into that, simplify the algebra and you're left with an expression in k.
     
    Last edited: Mar 16, 2006
  5. Mar 16, 2006 #4
    Why can't L'Hopital's rule work from the start?
     
  6. Mar 16, 2006 #5

    Curious3141

    User Avatar
    Homework Helper

    It can ! Stupid me. Orig poster, disregard my post and just differentiate numerator and denominator to get a single expression in x and set x = k.
     
  7. Mar 16, 2006 #6
    If you use L'Hopital rule in [tex] \frac {x^{1.5} - k^{1.5}}{x-k}[/tex], you will get [tex]\frac {6 \sqrt{k} -6 \sqrt{x}}{4 \sqrt{kx} *(x-k)'}[/tex]
    Then what can we do ?
     
    Last edited: Mar 16, 2006
  8. Mar 16, 2006 #7

    Curious3141

    User Avatar
    Homework Helper

    You're not differentiating correctly, the k is a constant and vanishes from both numerator and denominator.
     
  9. Mar 16, 2006 #8
    :bugeye: !!!!
    If only I knew when passing the test!! :frown:
    Thank you guys
     
    Last edited: Mar 16, 2006
  10. Mar 16, 2006 #9
    Please note that L'Hopital rule can only be used when your expression is equal to 0/0 or inf/inf. so you need to check your expression each time before you use the rule.
     
  11. Mar 23, 2006 #10

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    Multiply both the denominator and the numerator by [itex] x\sqrt{x}+k\sqrt{k} [/itex] and then simplify the fraction by [itex] x-k [/itex].


    Daniel.
     
  12. Mar 23, 2006 #11

    Galileo

    User Avatar
    Science Advisor
    Homework Helper

    Or change the variable [itex]u=x^4[/itex] and let [itex]a=k^4[/itex], then [itex]u \to a[/itex] as [itex]x \to k[/itex] and the limit becomes:

    [tex]\lim_{u \to a} \frac{u^{3/8}-a^{3/8}}{u-a}[/tex]
    which is the derivative of [itex]f(u)=u^{3/8}[/itex] at u=a.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: This was on my test
  1. Hypothesis Testing (Replies: 0)

  2. F-test / t-test (Replies: 37)

  3. Test Advice (Replies: 4)

  4. Hypothesis testing (Replies: 8)

  5. Hypothesis test (Replies: 6)

Loading...