The answer is quite simple, MS marches through the function between the plot limits. There is a discontinuity at x = 0, so as it moves from -epsilon to +epsilon, there is a large jump in the y value. The line there reflects this.
If you want to remove it, you can do the following:
The above will avoid the discontinuity.
You could even add the vertical asymptote, by adding a third plot line, if you so desire.
As to it not appearing in other functions, try tan(x). It will be there.
Lastly, I am puzzled by the 1.2 GB comment. That does not happen in OS X or IOS unless you have saved a great deal of scripts etc.