Question

What are the asymptotes and removable discontinuities, if any, of `f(x)=(4x^4 - x^5) / x`?

Answer 1

There is a hole at `x=0`

Explanation:

To begin, we must limit the function to ensure the denominator is not zero. The only excluded value in this case is `x != 0`. If we simplify under this assumption we get:

`(4x^4 - x^5)/x=4x^3-x^4,x!=0`

The resulting polynomial does not have any asymptotes, so only the hole at `x=0` from the original function remains.

`y=(4x^4 - x^5)/x`
graph{(4x^4-x^5)/x [-10, 10, -5, 5]}

Though it isn't visible on the graph, the hole is still there and must be listed in the excluded values.