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Math Help - Find the limit - unclear factorization

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    Find the limit - unclear factorization

    I have a function for which I have to find the limit (the solution is provided):

     \lim_{x \to -\infty} \frac{4x^{2} \sqrt{9x^{6} + 2x^{2}}}{x^{5} + 3} after evaluating the limit and proceeding to some factoring and simplifications I get  \lim_{x \to -\infty} \frac{4x^{2}|x^{3}| \sqrt{9+\frac{2}{x^{4}}}}{x^{5} + 3}

    What rule do we use to factor and to obtain this kind of absolute vale of x in this equation from under the square root with the subsequent changes there \frac{4x^{2}|x^{3}| \sqrt{9+\frac{2}{x^{4}}}}{x^{5} + 3} ?

    Can someone explain me this, and when it is applied?

    Thanks a lot,

    dokrbb
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    Re: Find the limit - unclear factorization

    Quote Originally Posted by dokrbb View Post
    I have a function for which I have to find the limit (the solution is provided):

     \lim_{x \to -\infty} \frac{4x^{2} \sqrt{9x^{6} + 2x^{2}}}{x^{5} + 3} after evaluating the limit and proceeding to some factoring and simplifications I get  \lim_{x \to -\infty} \frac{4x^{2}|x^{3}| \sqrt{9+\frac{2}{x^{4}}}}{x^{5} + 3}

    What rule do we use to factor and to obtain this kind of absolute vale of x in this equation from under the square root with the subsequent changes there \frac{4x^{2}|x^{3}| \sqrt{9+\frac{2}{x^{4}}}}{x^{5} + 3} ?

    Can someone explain me this, and when it is applied?

    Thanks a lot,

    dokrbb
    Since you are approaching \displaystyle -\infty, you can assume that \displaystyle x < 0 and so \displaystyle |x| = -x. Then

    \displaystyle \begin{align*} \lim_{x \to -\infty} \frac{4x^2 \left| x^3 \right| \sqrt{9 + \frac{2}{x^4}}}{x^5 + 3} &= \lim_{x \to -\infty} \frac{4x^2 \left( -x^3 \right) \sqrt{9 + \frac{2}{x^4}}}{x^5 + 3} \\ &= \lim_{x \to -\infty} \frac{-4x^5 \,\sqrt{9 + \frac{2}{x^4}}}{x^5 + 3} \\ &= \lim_{x \to -\infty}\frac{-4\,\sqrt{9 + \frac{2}{x^4}}}{1 + \frac{3}{x^5}} \\ &= \frac{-4 \, \sqrt{9 + 0}}{1 + 0} \\ &= -12 \end{align*}
    Thanks from dokrbb
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    Re: Find the limit - unclear factorization

    Quote Originally Posted by dokrbb View Post

     \lim_{x \to -\infty} \frac{4x^{2} \sqrt{9x^{6} + 2x^{2}}}{x^{5} + 3} after evaluating the limit and proceeding to some factoring and simplifications I get  \lim_{x \to -\infty} \frac{4x^{2}|x^{3}| \sqrt{9+\frac{2}{x^{4}}}}{x^{5} + 3}
    Thanks Prove It but I got that,

    I meant right at the beginning, how we obtain |x^{3}| at the numerator from the first equation, it's really this that I don't understand, sorry,

    can you show me that?
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    Re: Find the limit - unclear factorization

    Quote Originally Posted by dokrbb View Post
    Thanks Prove It but I got that,

    I meant right at the beginning, how we obtain |x^{3}| at the numerator from the first equation, it's really this that I don't understand, sorry,

    can you show me that?
    Never mind,

    I got the trick - it was a dumb question,

    thanks
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