Question

State the horizontal asymptote of the rational function. f(x)=5x+1/9x-2

Answer 1

hi, any ideas?

Answer 2

Not really no.

Answer 3

look at the degrees of the polynomials in the numerator and the denominator. what are they?

Answer 4

what is the polynomial in the numerator?

Answer 5

1

Answer 6

and the denominator?

Answer 7

The same

Answer 8

good. when that happens, the horizontal asymptote will be the line

Answer 9

\[y=\frac{\text{leading coefficient of numerator's polynomial} }{ \text{leading coefficient of denominator's polynomial} }\]

Answer 10

The line?

Answer 11

Oh so it will be 5/9?

Answer 12

yep! now that ONLY works if the numerator and denominator have the same degree polynomials

Answer 13

What happens if they don't?

Answer 14

then you get different situations...

if \[f(x) = \frac{ p(x) }{ q(x) }\] and the degree of p(x) is 1 more than the degree of q(x), you get a slant asymptote and it will be of the form y=ax+b where a \(\ne\) 0.

if the degree of q(x) is greater than p(x) then you'll get a horizontal asymptote of y = 0.

if f(x) has a different form, say \[f(x)=\frac{ p(x) }{ q(x) }+c\]where c is a non-zero constant and the degree of q(x) is less than p(x), then you get a horizontal asymptote of the form y = c.

Answer 15

I see. Thanks a lot for the help, I understand it now.

Answer 16

you're welcome!