Question

What are the asymptotes for `1/x`?

Answer 1

Have a look:

Explanation:

Here, for your function `y=1/x`, you have 2 types of asymptotes:

1) Vertical:
This is obtained looking at the point(s) of discontinuity of your function. These are problematic points where, basically, you cannot evaluate your function. In your case the point of coordinate `x=0` is one of these type of points. If you try using `x=0` into your function you get `y=1/0` which cannot be evaluated.
So the vertical line of equation `x=0`, the `y` axis, will be your VERTICAL ASYMPTOTE.

2) Horizontal.
This is a little bit more tricky...
You need to find a horizontal line towards which your function tends to get closer and closer.
One way to find this is to "see" what happens when `x` tends to become very big positively or negatively, i.e., `x->+-oo`.
You can see that, for `y=1/x`, when `x` becomes very big then `y` becomes very small....or tends to zero, `y->0`!!!
Basically, the `x` axis is your HORIZONTAL ASYMPTOTE!!!!

You can see these two asymptote graphically as the two lines near which the curve (representing your function) tends to get near to:
graph{1/x [-10, 10, -5, 5]}