Problem: Determine all asymptotes of
Solution: First we need to find the domain. The numerator exists everywhere and the denominator as well, so the only open question is whether the denominator can actually be equal to zero for some x. This very problem was studied here in Solved Problems - Using derivative to prove inequalities, and it turned out that the expression in the denominator is always positive. Thus the domain of the given function is the whole real line.
There are no proper endpoints in this domain and the given function is continuous everywhere, which means that there are no candidates for vertical asymptotes. Thus it remains to explore what happens at infinity and negative infinity, where we have a possibility of a horizontal or an oblique asymptote. The first step is to find limits there, if you need a hand with this, see e.g. Limits in Functions - Methods Survey.
There are two possible ways to handle this problem. First, we have a general version of l'Hospital's rule that also works for expressions of the type "something over infinity". Thus we do not have to worry about the outcome in the numerator, but still we would have to ask what happens in the denominator. It turns out that the indetermiate difference there gives infinity and l'Hospital's rule can be used.
How did we find out what happens in the denominator? Most likely by factoring out the dominant term there, but then it make sense to apply this method to the whole problem and factor out dominant terms in numerator and denominator. Calculations are somewhat nicer if we use substitution to change negative infinity to infinity.
We have to use l'Hospital anyway, namely to show that
Anyway, we have a proper limit at negative infinity, which means two things.
There is a horizontal asymptote given by
Now we look at what happens at infinity. There the situation is similar, again we will have a choice between l'Hospital rule and factoring out. We will show both ways, this time factoring out seems way easier (note that the exponential goes to zero now, therefore in the denominator we have a different dominant term).
Actually, here it would be easier to just cancel x in both numerator and denominator, try it. We found that the limit at infinity is improper. Thus there is no horizontal asymptote there, but there is a chance for an oblique asymptote. To determine this we first try to get A, see e.g. Asymptotes in Methods Survey - Graphing.
The limit converges, which means that there is still a chance for an oblique asymptote. It will be decided in the next limit, when we try to get B.
This limit converges, which confirms that there is an oblique asymptote at
infinity. Its equation is
