We explore functions that “shoot to infinity” at certain points in their domain.
Here we’ve made up a new term “slant” line, meaning a line whose slope is neither zero, nor is it undefined. Let’s do a quick review of the different types of asymptotes:
Vertical asymptotes Recall, a function has a vertical asymptote at if at least one of the following hold:
In this case, the asymptote is the vertical line
Horizontal asymptotes We have also seen that a function has a horizontal asymptote if and in this case, the asymptote is the horizontal line
Slant asymptotes On the other hand, a slant asymptote is a somewhat different beast.
To analytically find slant asymptotes, one must find the required information to determine a line:
- The slope.
- The -intercept.
While there are several ways to do this, we will give a method that is fairly general.
We are assuming these two limits are equal. Dividing by on the right hand side makes the limit equal to :
To find the value of , then, we can divide the left hand side by and evaluate the limit. We see the following.
So . We now know that for some value of . To find the -intercept , we use a similar method. Notice that so if we subtract from the right hand side, we are left with just . Since the two sides are equal, subtracting from the left hand side and evaluating the limit will give us the value for . We write the following.
By this method, we have determined that In other words, is a slant asymptote for our function . You should check that we get the same slant asymptote when we take the limit to negative infinity as well. We can confirm our results by looking at the graph of and :