Question

What are the vertical and horizontal asymptotes of f(x) = x/x+3-2 ?

Answer 1

See below.

Explanation:

Not quite sure of your notation here. I'm reading this as:

`f(x)=x/(x+3)-2`

Vertical asymptotes occur where the function is undefined.

For:

`x=-3`

`x/(0)-2`

Undefined, division by zero:

Vertical asymptote is the line:

`color(blue)(x=-3)`

We now see what happens as `x->+-oo`

`x/(x+3)-2`

Divide fraction by `x`:

`(x/x)/(x/x+3/x)-2`

Cancelling:

`1/(1+3/x)-2`

as `x->oocolor(white)(88888)` , `1/(1+3/x)-2=1/(1+0)-2=-1`

as `x->-oocolor(white)(88888)` , `1/(1+3/x)-2=1/(1+0)-2=-1`

This shows that as we approach `+-oo`, the function tends to `-1`.

This is a horizontal asymptote:

`color(blue)(y=-1)`

The graph of `f(x)=x/(x+3)-2` confirms this: