The horizontal asymptote in this function represents the value that the function approaches as x approaches positive or negative infinity. It is a visual representation of the long-term behavior of the function.
To determine the horizontal asymptote, we can use the quotient rule for limits. This means taking the limit as x approaches infinity of the function, which in this case would be 0. Therefore, the horizontal asymptote is y=0.
No, the horizontal asymptote of a function will remain the same unless the function itself is changed. It is a constant value that represents the long-term behavior of the function.
The bottom term of the function, (x4-81)1/2, does not affect the horizontal asymptote. It only affects the behavior of the function as x approaches infinity. The horizontal asymptote is determined by the overall behavior of the function, not just the bottom term.
Factoring the bottom term of the function can help us determine the behavior of the function as x approaches infinity. In this case, factoring the bottom term to (x2+9)(x2-9) can help us see that the function will approach y=0 as x approaches infinity, since both (x2+9) and (x2-9) will approach infinity at a slower rate than x4. This aligns with our previous determination of the horizontal asymptote.