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

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

To find the vertical, horizontal, and oblique asymptotes of the rational function T(x) = x^2 / (x^4 - 256), we need to analyze the behavior of the function as x approaches different values.

1. Vertical Asymptotes:
Vertical asymptotes occur when the denominator of the rational function becomes zero, but the numerator does not. In this case, the denominator is x^4 - 256. We can factor this as the difference of squares: (x^2 - 16)(x^2 + 16).

Setting the denominator equal to zero, we get:
(x^2 - 16)(x^2 + 16) = 0

This equation has two solutions:
x^2 - 16 = 0 => x^2 = 16 => x = ±4

So, the vertical asymptotes occur at x = -4 and x = 4.

2. Horizontal Asymptotes:
To find the horizontal asymptote, we look at the degrees of the numerator and the denominator.

The degree of the numerator is 2, and the degree of the denominator is 4. Since the degree of the denominator is greater, there is no horizontal asymptote. Instead, there is a slant or oblique asymptote (explained in the next step).

3. Oblique Asymptotes:
To find the oblique asymptote, we can perform long division or synthetic division to divide the numerator (x^2) by the denominator (x^4 - 256).

Performing synthetic division or long division, we get:
1
____________________
x^4 - 256 | x^2 + 0x + 0

x^2 - 256
__________
1

Therefore, the oblique asymptote is given by the equation y = x^2 - 256.

In summary:
- There are vertical asymptotes at x = -4 and x = 4.
- There is no horizontal asymptote.
- There is an oblique asymptote given by the equation y = x^2 - 256.