Final answer:
The function f(x) = x/(x^3 - 256) fails to exist when the denominator equals zero, which is at x = 6.3496. This value is a vertical asymptote of the function.
Step-by-step explanation:
The student is asking about the values for which the function f(x) = \frac{x}{x^3 - 256} fails to exist. This typically occurs at values of x that make the denominator equal to zero, since division by zero is undefined within the realm of real numbers.
First, we need to find when the denominator is zero:
x^3 - 256 = 0
x^3 = 256
x = \sqrt[3]{256}
x = 6.3496 (rounded to 4 decimal places)
Therefore, the function fails to exist when x = 6.3496, which is when the denominator x^3 - 256 equals zero resulting in a division by zero situation.
Understanding when functions are not defined due to asymptotes or limits, as mentioned in the reference information, is crucial in analyzing and graphing functions. The value x = 6.3496 represents a vertical asymptote for the function f(x).