Question

For what values of x, if any, does `f(x) = tan((5pi)/4-x)` have vertical asymptotes?

Answer 1

`f(x)` has vertical asymptotes at `....(3pi)/4,-pi/4,-(5pi)/4,...`

We can say that `f(x)` is undefined for any applicable `(3pi)/4+npi`, where `ninZZ`.

Explanation:

`tan(x)` is undefined at certain points:

`pi/2,(3pi)/2,(5pi)/2`, and so on.

Graphing `tan(x),` we find that, where `h` is a point where `tan(x)` is undefined:

`lim_(xrarrh^+)tan(x)=oo`, and `lim_(xrarrh^-)tan(x)=-oo`

We'll find the first three vertical asymptotes.

So here, `f(x) " DNE"` when `((5pi)/4-x)rarrh`.

Taking `h=pi/2`, we have:

`((5pi)/4-x)rarrpi/2`, which we can write as:

`(5pi)/4-x=pi/2`

`-x=pi/2-(5pi)/4`

`-x=-(3pi)/4`

`x=(3pi)/4`

Taking `h=(3pi)/2`, we have:

`(5pi)/4-x=(3pi)/2`

`-x=(3pi)/2-(5pi)/4`

`-x=pi/4`

`x=-pi/4`

For `h=(5pi)/2`, we have:

`(5pi)/4-x=(5pi)/2`

`-x=(5pi)/2-(5pi)/4`

`-x=(5pi)/4`

`x=-(5pi)/4`

This series continues to the left and right.