## owlet one year ago Question below:

1. owlet

2. owlet

what's the next step?

3. owlet

@SolomonZelman @Loser66 @freckles

4. SolomonZelman

$$\Large\color{black}{\displaystyle\lim_{x \to~ -\infty}\frac{\sqrt{2x^2+x+1}}{3x-1}}$$

5. SolomonZelman

you can do this: $$\Large\color{black}{\displaystyle\lim_{x \to~ -\infty}\frac{\sqrt{2x^2+x+1\color{white}{\LARGE |}}}{3x-1}}$$ $$\Large\color{black}{\displaystyle\lim_{x \to~ -\infty}\frac{\sqrt{2x^2+x+1\color{white}{\LARGE |}}}{\sqrt{(3x-1)^2}}}$$ $$\Large\color{black}{\displaystyle\lim_{x \to~ -\infty}\sqrt{\frac{2x^2+x+1}{(3x-1)^2}}}$$

6. SolomonZelman

$$\Large\color{black}{\displaystyle\lim_{x \to~ -\infty}\sqrt{\frac{2x^2+x+1}{9x^2-6x+1}}=\sqrt{2/9}=\sqrt{2}/3}$$

7. SolomonZelman

you can bring the limit inside the root, and do L'Hospital's rule twice.

8. owlet

how will i apply x approaches -infinity though?

9. SolomonZelman

Ok, $$\large\color{black}{\displaystyle\lim_{x \to~ -\infty}\sqrt{\frac{2x^2+x+1}{9x^2-6x+1}}=\sqrt{\lim_{x \to~ -\infty}\frac{2x^2+x+1}{9x^2-6x+1}}}$$

10. SolomonZelman

$$\large\color{black}{\displaystyle \sqrt{\lim_{x \to~ -\infty}\frac{2x^2+x+1}{9x^2-6x+1}}}$$ You could apply L'Hospital's rule at this point, because as x$$\to -\infty$$, the top and bottom tend to ∞. (an ∞/∞ case)

11. SolomonZelman

differentiate on top and bottom and you get $$\large\color{black}{\displaystyle \sqrt{\lim_{x \to~ -\infty}\frac{4x+1}{18x-6}}}$$

12. owlet

but we haven't learned l'hospital's rule yet

13. SolomonZelman

Oh, so you aren't allowed to use it? (But do you know that rule, though?)

14. owlet

no, we're not allowed and I don't know that rule also

15. SolomonZelman

$$\large\color{black}{\displaystyle \sqrt{\lim_{x \to~ -\infty}\frac{2x^2+x+1}{9x^2-6x+1}}}$$ You can just conclude the answer based on the leading coefficients, then. (The limit will be equal to 2/9 and the square root of 2/9 is going to be equal to √2 /3

16. SolomonZelman

It really did seem like an L'Hospital's rule kind of a problem, I apolgoize for mentioning it.

17. owlet

sorry i was off. I just finished my class. will it always work though? for those kind of problems? Just get the leading coefficients? I want to learn that rule.. where can I learn it? maybe I can just search it up on the internet.