Question

Limit question here, I'm confused on how to get the answer:

Answer 1

\[\lim \sqrt{9x ^{2}+ x} - 3x \] as x -> infinity

Answer 2

I tried rationalizing the numerator and got:
\[ \frac{ 9x ^{2} + 8x }{ \sqrt{9x ^{2}+x}+3x}\]

Answer 3

and this is where i'm confused. I'm not quite sure what to do next. I know it has something to do with leading coefficients but i'm really confused.

Answer 4

numerator?

Answer 5

What do you mean?

Answer 6

Sorry, now i know what you meant. I tried putting the original problem into a fraction then rationalizing the numerator

Answer 7

Was that a bad move?

Answer 8

it might not be a bad move, but I cant really determine what the original problem was that needs its limit taken.

you posted:\[\lim \sqrt{9x ^{2}+ x} - 3x\]
this has no fraction in it to denote a numerator to rationalize.

Answer 9

the conjugate of what you wrote is\[\sqrt{9x ^{2}+ x} + 3x\]
and when multiplied we get\[{9x ^{2}+ x} - 9x^2=x\]