Find the vertical, horizontal, and oblique asymptotes, if any, for the following rational function.

T(x)=x^2/x^4-256

x^4-256 = (x^2+16)(x+4)(x-4)

so,
vertical at x=4,-4
horizontal at y=0
no oblique

To find the vertical, horizontal, and oblique asymptotes for the rational function T(x) = x^2 / (x^4 - 256), we need to analyze the behavior of the function as it approaches infinity and as it approaches certain specific points.

1. Vertical Asymptotes:
Vertical asymptotes occur when the function approaches infinity or negative infinity as x approaches a certain value. In this case, we need to find the values of x that make the denominator of the rational function equal to zero.

(x^4 - 256) = 0
(x^2 + 16)(x^2 - 16) = 0

Setting each factor equal to zero gives us two potential values for x: x = -4 and x = 4.

Therefore, there are two vertical asymptotes, which are vertical lines at x = -4 and x = 4.

2. Horizontal Asymptotes:
To find the horizontal asymptotes, we need to check the degrees of the numerator and denominator of the rational function.

The degree of the numerator is 2, and the degree of the denominator is 4. When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is located at y = 0.

Therefore, there is a horizontal asymptote at y = 0.

3. Oblique Asymptotes:
To check for oblique (slant) asymptotes, we need to divide the numerator by the denominator using polynomial long division.

x^2
____________
x^4 - 256 | x^2

- (x^4 - 256)
_____________
256

The quotient is x^2 / (x^4 - 256) with a remainder of 256. Since there is a non-zero remainder, the function does not have an oblique asymptote.

In summary:
- The function T(x) = x^2 / (x^4 - 256) has vertical asymptotes at x = -4 and x = 4.
- It has a horizontal asymptote at y = 0.
- There are no oblique asymptotes.