Determine the vertical and
horizontal asymptotes of the function 𝑓 of 𝑥 equals negative one plus three over
𝑥 minus four over 𝑥 squared.
Let’s start with the vertical
asymptotes. Vertical asymptotes are vertical
lines where the denominator in a rational function equals zero. In this function, we see that we
have 𝑥 and 𝑥 squared in the denominator. And that means 𝑥 cannot be equal
to zero. So we say that the vertical
asymptote of this function is at 𝑥 equals zero. The function doesn’t exist at 𝑥
Now horizontal asymptotes are a
little bit different. Horizontal asymptotes indicate
general behaviour usually far off to sides of the graph. And when we look for horizontal
asymptotes, we need to consider a few cases. We have the case that the degree of
the denominator is larger than the degree of the numerator. And the case where the degree of
the numerator is equal to the degree of the denominator. For now, we won’t consider the
cases where the degree of the numerator is larger than the degree of the
At first glance, it might seem like
the degree of the numerator here is zero because there’re no 𝑥-values in the
numerator. But we need to see the case when we
add all three of these terms together. We need a common denominator. If we use 𝑥 squared as our common
denominator, we can rewrite negative one as negative 𝑥 squared over 𝑥 squared.
And then, to convert three over 𝑥
into a fraction with the denominator of 𝑥 squared or multiply three over 𝑥 by 𝑥
over 𝑥. And that becomes three 𝑥 over 𝑥
squared. And the four over 𝑥 squared
doesn’t change. And we’ve rewritten our function to
look like this, negative 𝑥 squared plus three 𝑥 minus four all over 𝑥
squared. The degree, the highest exponent of
our numerator, is two. And the degree of our denominator,
the highest exponent in the denominator, is also two. So we have the second case. The degree in our numerator equals
the degree in our denominator.
When this is the case, there is an
horizontal asymptote at the place 𝑦 equals the numerator’s leading coefficient over
the denominator’s leading coefficient. The leading coefficient in our
numerator is negative one. And the leading coefficient in our
denominator is positive one. Therefore, we did have a horizontal
asymptote at negative one over one or, more simply, at 𝑦 equals negative one.
Our vertical asymptote is at 𝑥
equals zero. And our horizontal asymptote is at
𝑦 equals negative one.