Question

determine the horizontal asymptote if one exists

Answer 1

Whats the function?

Answer 2

\[f(x)= \frac{ -8(x-4)^{3}(x+6)^{6} }{ x ^{9}+10x ^{3} +24}\]

Answer 3

wow thats a mighty function well there are multiple ways of doing this. We could take the limit as the function goes to infinity or negative infinity and compare the values we get. Expand the top and see if we get a ratio of powers or use a method call Lhopitals Rule which is taking derivatives of top and bottom until the function doesnt get an indeterminant answer anymore

All depends on if you know calculus very well or not. IF you do don't i suggest we do the expand and compare method

Answer 4

yeah i know calculus, which way would be the easiest you think?

Answer 5

probably taking the derivatives?

Answer 6

Lhopitals rule would be a pain since you need to do it 10 times to the bottom in order to get a straight number without variables.

The limit method is the same way pain to lug infinity into there so my suggestion is we expand the top then see if the degree of the numerator is bigger, the degree of the denominator is bigger, or they are equal

Answer 7

or the cheap way and graph it on a graphing calculator always works

Answer 8

ok, let me try graphing it then

Answer 9

Wait. just realized how to do this without a calculator

Answer 10

lol okay, how?

Answer 11

well if you look at just the portions at the top if you cube that first part your leading term would be x^3 right? and then the second part you expand it to the sixth power your leading term is x^6 and if you were to multiple all of that junk together you would get x^9 as the leading term and now multiply x^9 by the -8 in the top and you have -8x^9 and since you have x^9 in the bottom as well your numerator and denominator degrees are equal and you can do a simple ration. Your horizontal asymptotote is -8

Answer 12

thanks :)

Answer 13

your welcome

Answer 14

can you help me with another problem?

Answer 15

sure