Question

Give an example of a rational function that has no horizontal asymptote and a vertical asymptote at
x = 1.

Answer 1

@amistre64 @phi @mathstudent55

Answer 2

what are some basic rules that you can apply here?

Answer 3

Im really not sure. I was going to approach this with guess and check but I figured you could help me approach it

Answer 4

there are 3 fundamental rules for HA calculations

when the degree of the top is _________ the degree of the bottom, the HA is ______

Answer 5

degrees can only be greater than, equal to, or less than .... each case gives us a specific HA assessment

Answer 6

Okay. Im not positive as to what you mean

Answer 7

your material for your course should define 3 cases for rational function

Answer 8

by comparing the degree of the top and bottom polynomials, the horizontal asymptote can be determined

Answer 9

Okay. So is the a formula of some type that shows me the relation of the two?

Answer 10

recall that a HA is only defined for very large values of x therefore the leading terms of the top and bottom polys take control of the values\[\frac{ax^n}{bx^m}\]

Answer 11

if n=m, then for large values of x, the asymptote is a/b

when n<m then the bottom gets bigger quicker than the top and we go to zero

when n>m then the top get bigger quicker and we ahve no HA

Answer 12

any of this not make sense?

Answer 13

I think I follow

Answer 14

compare:
\[\frac{2x^3}{3x^3}\]
x^3/x^3=1 so, HA = 2/3


\[\frac{2x}{3x^3}\]
10000/10000^3 so we have a very small value that approaches zero



\[\frac{2x^3}{3x}\]
10000^3/10000 this is just a bigger and bigger number

Answer 15

we know what the bottom needs to be right?

Answer 16

bottom = 2/3?

Answer 17

what defines a vertical asymptote?

Answer 18

The distance of the curve from the x axis

Answer 19

a vertical asymptote is a factor that cannot be removed ....

\[\frac{something}{x-1}\]
as long as something doesnt have x-1 as a factor, then the bottom cant be removed and we have a VA at x=1

Answer 20

now, in order for the thing to have no HA, the top has to be a poly of greater degree than the bottom. any ideas for the top?

Answer 21

x^2?

Answer 22

x^2 is fine :)

Answer 23

Okay Great :)

Answer 24

So what next?

Answer 25

submit the rational expression of course

Answer 26

So its just x^2/x-2

Answer 27

I mean x-1

Answer 28

x-1 but yes

Answer 29

Oh wow! That was simpler than I thought. Thanks so much!!

Answer 30

yw