We discuss the circumstances that generate vertical asymptotes in rational functions.
We have encountered vertical asymptotes in the context of certain function types, but rational functions are special in that they give a way to generate multiple vertical asymptotes. Moreover, their location and functional behavior around the vertical asymptotes is not always the same. For this reason it warrants special attention. Unfortunately to properly study the nature of domain restrictions one needs limits, which we won’t introduce until calculus. Despite this, there are some general guidelines we can use that will work so long as the functions in the numerator and denominator are continuous functions (independently) near the domain restrictions.
In general, assuming the continuity condition mentioned above, if an value resolves to zero in the denominator function, and non-zero in the numerator function, then there is a vertical asymptote at that value. This is true even in a simplified version of the rational function.
The first thing we want to do to solve this problem is to factor the numerator and denominator if possible. Doing so gets us . This gives us domain restrictions of and .
At this point we could check both these values. If we do; we get the numerator and denominator are both for and we get the numerator is and the denominator is when . From this we know that there is a vertical asymptote at . However, something of the form “” it is called an indeterminate form and we don’t know what is happening at this value without doing further work.
Ideally we hope to simplify the function to resolve our indeterminate issue. Since we were able to factor the top and bottom, we can simplify the fraction down to . Now if we check we end up with , which is not “non-zero over zero” and thus is not a vertical asymptote (indeed, it is a hole in the graph, which we will discuss in the next section).
So, we can conclude that the only vertical asymptote that we know exists for sure is at .
Now lets consider a more complicated example.
The takeaway is that, under certain continuity conditions near the domain restriction, if you try to evaluate a value and get something of the form “non-zero over zero” that domain restriction represents a vertical asymptote. If you still get something of the form “zero over zero” then it is indeterminate and you need better tools (limits) to determine what is happening. That leaves one more situation; “something over non-zero”. In this last case, we have a hole at the domain restriction, and we discuss that next.