What software that properly calculate asymptote? I am pretty sure I can ask about software that mathematicians use.
I have been exploring offline and online software to make a simple graph :

$$y=3^x - 1$$

However most of them crossed the asymptotes when it isn't supposed to.
because $y\ne -1$. If $-1=3^x - 1$, then  $3^x=0$, which is never true.
So when I generate the graph, it will always result in some points to be $y=-1$.

So which software(s) that can properly avoid cutting asymptote $(y=-1)$

Software I have tried:


*

*Microsoft Mathematics

*Some cheapo online tools that come first in google search(es).

*Famous online graphing tool, Desmos, which gives an unwanted result:

 A: The number $3^{-8.83}-1$ is within $6\times 10^{-5}$ of $-1$.  The scale of your image looks like about $1$ centimeter per unit.  So to separate the graph of the function from the asymptote, a resolution on the other of $10{,}000$ pixels per centimeter, or $25{,}400$ pixels per inch, is necessary.
Wikipedia says the spatial resolution of standard computer monitors is $72$–$100$ pixels per inch.
A: @MatthewLeingang's accepted answer settles the mathematical question you asked.
In your comment you add

I need the thing that shows the lil bit floating and doesnt return the
  value y=-1 in the curve.

If all you want is the picture to make your point, consider asking the software not to show the point $(-8.83, -1)$ as the red dot, or edit it out in a tool like photoshop. Then your reader will see the graph overlaying the axis (because it's so close) and the little uptick at the end and not be confused.
A: The idea of vector graphics is that if you zoom in on an image,
you continue to get the best image your monitor can display
(or the best your printer can print),
unlike raster graphics where you just get bigger dots.
Using a graphing tool such as Desmos, you can zoom in on any particular
part of your graph in order to see fine details such as the very
tiny separation between the graphs of $y=3^x - 1$ and $y = -1$
at $x = -8.83$. Here's an image of Desmos.com zoomed in on that
section of the graph:

The distance between the thick lines is $0.001$ in this figure
(see the coordinates along the top and left edges of the figure)
and the distance between the thin lines is $0.0002$.
The origin $(0,0)$ would be a very large distance to the upper right
if we had a large enough screen to show it on.
The downside of zooming in like this is that you can't see the
general shape of the graph, since you are looking at such a tiny 
portion of it. To better visualize the graph you might want to look at 
several screenshots at different zoom levels.
