How do you identify all asymptotes or holes for #f(x)=(x^2+2x-8)/(-4x)#?

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

Slant asymptotes occur when the degree of the numerator > degree of the numerator. This is the case here ( numerator-degree 2, denominator-degree 1 ) Hence there is a slant asymptote.

Holes occur when there is a duplicate factor on the numerator/denominator. This is not the case here hence there are no holes. graph{(x^2+2x-8)/(-4x) [-10, 10, -5, 5]}

To identify all asymptotes or holes for the function ( f(x) = \frac{x^2 + 2x - 8}{-4x} ):

1. Find any vertical asymptotes: Vertical asymptotes occur where the denominator equals zero, except where the numerator also equals zero, which would indicate a hole. Solve the equation ( -4x = 0 ) to find the vertical asymptote.

2. Determine if there are any holes: If there are any common factors in both the numerator and the denominator, these would indicate potential holes in the graph. Factor both the numerator and the denominator completely and simplify, looking for common factors.

3. Horizontal or slant asymptotes: Determine if the degree of the numerator is less than, equal to, or greater than the degree of the denominator. Depending on this relationship, there may be a horizontal asymptote, a slant asymptote, or neither.

4. Check for any oblique asymptotes: If the degree of the numerator is exactly one more than the degree of the denominator, there may be an oblique (slant) asymptote.

Apply these steps to identify all asymptotes or holes for the given function.