Question

What are the asymptote(s) and hole(s), if any, of `f(x) =sin(pix)/x`?

Answer 1

Hole at `x=0` and a horizontal asymptote with `y = 0`

Explanation:

First you have to calculate the zero marks of the denominator which in this case is `x` therefore there's a vertical asymptote or a hole at `x = 0`. We aren't sure whether this is a hole or asymptote so we have to calculate the zero marks of the numerator
`<=> sin(pi x) =0`
`<=>pi x = 0 or pi x = pi`
`<=> x = 0 or x = 1`
As you see we have a common zero mark. This means that it's not an asymptote but a hole (with `x=0`) and because `x=0` was the only zero mark of the denominator that means that they're are no vertical asymptotes.

Now we take the `x`-value with highest exponent of the denominator and of the numerator and divide them by each other.
but because there is only one kind of exponent of `x` , the function `f(x)` doesn't change.
`<=> sin(pi x)/x`
Now, if the exponent is bigger in the numerator than of the denominator that means that there's a diagonal or a curved asymptote. Otherwise, there's a straight line. In this case, it's going to be a straight line. Now you divide the a values of the numerator by the a value of the denominator.
`<=> Sin(pi)/1`
`<=> 0/1`
`<=> 0`
`<=> y = 0` `=` the horizontal asymptote