Question

How do you find/know where/when the graph crosses the horizontal asymptote?

Answer 1

do you have a specific example, because this is a bit vague.
if you know the equation for the horizontal asymptote you can set the function equal to that number and solve

Answer 2

I know, I am sorry, but I thought it was a general response... will the graph only cross the horizontal asymptote when n=k, i.e. the degree of the numerator is the same as the degree of the denominator

For example, n=2, k=2

\[f(x)=\frac{x^2-3x-4}{x^2+x-6}\]

Answer 3

well no, for example \(\frac{\sin(x)}{x}\) has a horizontal asymptote at \(y=0\) and it crosses it infinitely often

Answer 4

the horizontal asymptote in your example is \(y=1\) since the degree of the numerator and denominator are the same (they are both two) and the leading coefficient is 1 in both top and bottom

Answer 5

if you want to see if it crosses it at all, you would have to set it equal to one and solve, i.e. solve
\[x^2-3x-4=x^2+x-6\]

Answer 6

looks like we get
\[-3x-4=x-6\]
\[-4=4x-6\]
\[2=4x\]
\[x=\frac{1}{2}\]

Answer 7

@satellite73 So what would we have to get to signify that it wont cross the horizontal asymptote?

Answer 8

if it's imaginary? or no solution?

Answer 9

Guide to asymptotes:
A rational expression has three types of asymptotes. There are vertical, horizontal, and oblique/slant asymptotes.
Vertical Asymptotes: Vertical Asymptotes occur when the denominator of a rational expression equals 0. This means that for any x-value in which the denominator becomes 0 will form a vertical asymptote as x = ? where the question mark is replaced by the x-value. However, there is always a chance for discontinuities or "holes" is the graph. THis occurs when the denominator becomes 0. Now, the difference is that the cause of the "hole" can be canceled. FOr instance, in the following example, if you have:\[\frac{(x + 3)(x + 2)}{(x + 2)(x - 5)}\]the value x = -2 forms a hole rather than a vertical asymptote.
Horizontal Asymptotes:
There are two instances of horizontal asymptotes. If the degree of the numerator is equal to the degree of the denominator, take the ratio of the leading coefficients. For example, in your problem, the leading coefficients are each 1 because the x^2 term is the term with the biggest degree. The ratio is 1:1 or 1/1 which is 1. Therefore, the horizontal asymptote is at y = 1. THe second instance of a horizontal asymptote occurs when the degree of the denominator is bigger than the numerator. If this happens, it's ALWAYS AT y = 0!
Oblique/Slant Asymptotes:
These occur when the numerator has a higher degree than the denominator, but only by 1 degree. You find the equation of this by using long division and taking the quotient without the remainder. If the quotient is x + 5 + 3/(x - 5) where 3(x - 5) is the remainder, the equation is y = x + 5.

Answer 10

how do you know that it forms a hole rather than a vertical asymptote?

Answer 11

It's just a fact. Since it's not a vertical asymptote, there can be another value where it crosses that value.

Answer 12

I don't get it...