Question

Why cant a graph have more than two horizontal asymptotes

Answer 1

Go to the definition of horizontal asymptote and the definition of function.

Horizontal asymptotes indicate what happens to a function when x gets really really large, either in the negative or positive direction.

Take the positive direction. A function can have only one y-value for any x value and as x gets larger and larger, it therefore can have only one value to stay true to the definition of a function.

The same is true for the negative direction. For large negative values, it is the coefficients that determine the sign, not the variable x itself, because for negative over negative is positive.

for example (2x + 1) / x approaches 2 as x gets larger and large and has as gets more and more negative (towards negative infinity) the function approaches 2 as well. So regardless of the sign of x, it is the coefficients that determine the sign.

Note that if this is a horizontal asymptote, that the long-term value of x in the positive direction must equal the long-term value of x in the negative direction, otherwise, the asymptote would NOT be horizontal.