How do you find the asymptotes for #y= (14x)/(x^4 +1)^(1/4)#?

To find the asymptotes for ( y = \frac{14x}{(x^4 + 1)^{\frac{1}{4}}} ), we need to consider both horizontal and vertical asymptotes.

Horizontal asymptotes can be found by examining the behavior of the function as ( x ) approaches positive or negative infinity. For this function, as ( x ) approaches positive or negative infinity, the denominator ( (x^4 + 1)^{\frac{1}{4}} ) grows without bound, making the entire expression approach zero. Therefore, the horizontal asymptote is ( y = 0 ).

Vertical asymptotes occur where the function is undefined, typically when the denominator of a rational function equals zero. In this case, ( (x^4 + 1)^{\frac{1}{4}} ) cannot be zero, so there are no vertical asymptotes for this function.

Therefore, the only asymptote for the function ( y = \frac{14x}{(x^4 + 1)^{\frac{1}{4}}} ) is the horizontal asymptote ( y = 0 ).