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Math Help - Limit Calculation

  1. #1
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    Limit Calculation

    Limit Calculation-lim.jpg

    please help me with this one to
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  2. #2
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    Re: Limit Calculation

    If x\rightarrow 2, then 5x-10\rightarrow0 and \sqrt[3]{x-1}-1\rightarrow0. So we have an indeterminate form, \frac{0}{0}, use l'H˘pital's rule.

    The answer is 15.
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  3. #3
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    Re: Limit Calculation

    A standard way to find limits of the form \lim_{x\to a}\frac{f(x)-c^2}{\sqrt{f(x)}-c} where \lim_{x\to a}f(x)=c^2 is to multiply the numerator and the denominator by \sqrt{f(x)}+c to get

    \frac{(f(x)-c^2)(\sqrt{f(x)}+c)}{(\sqrt{f(x)}-c)(\sqrt{f(x)}+c)} = \frac{(f(x)-c^2)(\sqrt{f(x)}+c)}{f(x)-c^2} = \sqrt{f(x)}+c

    where the latter expression does not have a singularity at a.

    For this problem, use the same idea and the fact that

    a^3-b^3=(a-b)(a^2+ab+b^2) (*)

    Equivalently, factor out 5 and represent x - 2 in the numerator as (x - 1) - 1 and then factor it according to (*).
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  4. #4
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    Re: Limit Calculation

    Thank you very much now i get it
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