What are the asymptotes for #1/x#?

Question

Cora Alexander · Accepted Answer

Have a look:

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]}

Liam Brennan · Answer

undefined The function $ \frac{1}{x} $ has two types of asymptotes: vertical asymptotes and horizontal asymptotes.

1. Vertical Asymptotes:
   Vertical asymptotes occur where the function approaches positive or negative infinity as $ x $ approaches a certain value. For $ \frac{1}{x} $, vertical asymptotes occur where the denominator $ x $ equals zero. Therefore, the vertical asymptote for $ \frac{1}{x} $ is the line $ x = 0 $.

2. Horizontal Asymptotes:
   Horizontal asymptotes occur when the function approaches a constant value as $ x $ approaches positive or negative infinity. For $ \frac{1}{x} $, as $ x $ approaches positive or negative infinity, the function approaches zero. Therefore, the horizontal asymptote for $ \frac{1}{x} $ is the line $ y = 0 $ or the x-axis.

In summary:
- Vertical asymptote: $ x = 0 $
- Horizontal asymptote: $ y = 0 $ or the x-axis.