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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


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    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


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    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


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    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


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    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].

  12. Mar 23, 2006 #11


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    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.
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