Thread: Find the limit - unclear factorization

1. 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

2. Re: Find the limit - unclear factorization

Originally Posted by dokrbb
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*}

3. Re: Find the limit - unclear factorization

Originally Posted by dokrbb

$\lim_{x \to -\infty} \frac{4x^{2} \sqrt{9x^{6} + 2x^{2}}}{x^{5} + 3}$ 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?

4. Re: Find the limit - unclear factorization

Originally Posted by dokrbb
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