**Can Horizontal Asymptotes Be Crossed**. The function $f(x)=\sin x/(x^2+1)$ crosses its asymptote $y=0$ infinitely many times. $\begingroup$ this idea that the graph cannot cross a horizontal asymptote is a myth.

Horizontal asymptotes exist for functions where both the numerator and denominator are polynomials. There is nothing in this definition that requires that the asymptote cannot be crossed. Horizontal asymptote when [latex]f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)},q\left(x\right)\ne 0\text{where degree of }p=\text{degree of }q[/latex].

### Notice That, While The Graph Of A Rational Function Will Never Cross A Vertical Asymptote, The Graph May Or May Not Cross A Horizontal Or Slant Asymptote.

As andrei pointed out, it is false even for rational functions. Note that this is not the case with any vertical asymptote as a vertical asymptote never intersects the curve. Horizontal asymptotes exist for features in which each the numerator and denominator are polynomials.

### Horizontal Asymptotes Exist For Functions Where Both The Numerator And Denominator Are Polynomials.

The feature can contact or even move over the asymptote. Here is a simple graphical example where the graphed function approaches, but never quite reaches, \(y=0\). The function can touch and even cross over the asymptote.

### You Didn't Read The Above Carefully Enough.

Horizontal asymptotes describe the behavior of a graph for large or very. It’s possible to have multiple crossings. However, i should point out that horizontal asymptotes may only appear in one direction, and may be crossed at small values of x.

### However, I Should Point Out That Horizontal Asymptotes May Only Appear In One Direction, And.

Horizontal asymptotes exist for features in which each. Whereas you can never touch a vertical asymptote, you can (and often do) touch and even cross horizontal asymptotes. Of course, it will depend on the degrees of the polynomials, and the computations needed to find them might get nasty.

### When Dealing With Rational Functions (Which Is Presumably What You Are Doing), What You Can Say Is That After A Certain Point The Function Will No Longer Cross The Horizontal Asymptote And Will Just Approach It.

Horizontal asymptote y=0 y = 0 when f(x)= p(x) q(x),q(x)≠0 where degree of p<degree of q f ( x) = p ( x) q ( x), q ( x) ≠ 0 where degree of p < degree of q. I.e., there may exist a value of x such that f(x) = k. Functions are regularly graphed to offer a visual.