Question

Which of the following would have a horizontal asymptote of y = b/a? Which one has a horizontal asymptote of y = 0?

( bx^3 - x^2 + 3 ) / ( ax^2 - 2 )

( bx^3 - x^2 + 3 ) / ( ax^3 - 2 )

( bx^3 - x^2 + 3 ) / ( ax^4 - 2 )

( bx^2 / a ) - 6

Please explain! :)

Answer 1

b/a is the same as bx^n/ax^n, so that would be

( bx^3 - x^2 + 3 ) / ( ax^3 - 2 )

as x gets large, the lower powers become insignificant, so you just have to worry about the highest power of top and bottom.

If the top has lower power than bottom, the asymptote is always y=0. That would be the 3rd function

If the top has a power one more than the bottom, there will be a slant asymptote. The function is basically bx/a + c

Answer 2

Okay, I think I got it... So then, of these, the limit as x approaches infinity does not exist for the second one?

Answer 3

ummm. The second one is the answer to the question posed. The limit is b/a.

The limit does not exist when the top power is greater than the bottom power. That would be the first one. The limit is bx/a, which grows without limit as x grows. The 4th one also grows unbounded.