find all vertical and horizontal asymptoles

f(x)=4x/x^+1

Horizontal asymptotes occur when the numerator is equal to 0, vertical when denominator is equal to 0

therefore, in this problem, both asymptotes are 0.

To find the vertical and horizontal asymptotes of the function f(x) = 4x / (x^2 + 1), follow these steps:

1. Vertical Asymptotes:
Vertical asymptotes occur when the denominator of a rational function is equal to zero, but the numerator is not zero. In this case, there are no vertical asymptotes because the denominator, x^2 + 1, is never equal to zero for any real value of x.

2. Horizontal Asymptotes:
Horizontal asymptotes occur when the degree of the numerator and denominator of a rational function is the same. We can determine the horizontal asymptote(s) by comparing the degrees of the numerator and denominator.

The degree of the numerator of the function f(x) is 1 (since the highest power of x is x^1), and the degree of the denominator is 2 (since the highest power of x is x^2). Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is at y = 0.

So, the horizontal asymptote for the function f(x) = 4x / (x^2 + 1) is y = 0.