Question

lim(x->infinity) ( sqrt(7+9x^2))/(1-2x). Am i supposed to rationalize the numerator?

Answer 1

It's gonna be -3/2.

Answer 2

In the limit, the 9x^2 dominates, so the numerator is effectively 3x.

Answer 3

Likewise, denom -2x dominates.

Answer 4

So 3/-2, or -3/2 is the lim.

Answer 5

you basically ignore numbers that have little affect on the answer for example you ignore 7

Answer 6

so all you have is sqrt ( 9x^2)/-2x

Answer 7

3x/-2x = -3/2

Answer 8

oh is that just a rule? the ration between the two factors with the biggest exponent?

Answer 9

sound about right.

Answer 10

well that explains a lot aha

Answer 11

Be careful. You really have to look at all the terms. The art is in recognizing which terms will dominate close to the limit ... often pretty easy for "really big" x, when the lim is x -> infty.

Answer 12

if it was goign to negative infinity would it change anything?

Answer 13

Just the sign.

Answer 14

the terms dominating would be the term with the biggest postive exponent, and the term with the biggest negative exponent?

Answer 15

the any term with a negative x in the exponent will go to zero in the limit as x goes to positive infinity

Answer 16

\[\lim_{x\to\infty}a^{-x}=0\]

Answer 17

but when it goes to negative infinity?

Answer 18

then the exponent approaches positive infinity, so the limit is positive infinity\[\lim_{x\to-\infty}a^{-x}=\infty\]

Answer 19

oh! okay thanks

Answer 20

welcome :D

Answer 21

i feel like sometimes the way its taught is so confusing, when theres way easier ways to figure it out haha