Question

Question: Take the limit to infinity of (3x-5x^2)/(2x^2 +4)

Answer 1

Is it -infinity/infinity?

Answer 2

that's the limit as x -> infinity?

Answer 3

yes

Answer 4

lim x->inf

Answer 5

no, then with the highest exponent of x in both numerator and denominator being equal, it becomes just the ratio of their coefficients

Answer 6

you agree that the behavior of the term with the largest exponent dominates the behavior as x->infinity, right?

Answer 7

yes, I remember that, just not sure how to carry it out

Answer 8

okay, so "simplify" to \[\frac{-5x^2}{2x^2}\]what's that going to do as \(x\rightarrow\infty\)?

Answer 9

a negative number

Answer 10

it's going to have the same value it has at any non-zero value of \(x\), which is...

Answer 11

-5/2 ?

Answer 12

indeed

Answer 13

ah ok, so when I'm faced with these problems

Answer 14

drop everything but the highest power of x in the numerator and denominator

Answer 15

then determine it's simplest form?

Answer 16

if the numerator has a term that has a higher exponent, then you just find the limit of that term.

\[\lim_{ x\rightarrow\infty} \frac{x^2}{3x} = \infty\]because there's nothing "holding it back"

\[\lim_{x\rightarrow\infty}\frac{x}{3x^2} = 0\]

it's really only the case where the highest exponent in both numerator and denominator are equal that anything interesting happens...

Answer 17

ah ok

Answer 18

Yeah, it's just like finding the end behavior of a polynomial. Only the highest exponent term has any real impact once you move away from 0.

Answer 19

One of the relatively few times you get to just throw stuff away and still get the right answer :-)

Answer 20

XD thanks again

Answer 21

you bet