Question

How do you identify the vertical and horizontal asymptotes for rational functions?

Answer 1

Hint: According to the question, first we will find the vertical asymptotes, and then we will find the horizontal asymptotes. For vertical asymptotes, we will cancel out the common terms from both the numerator and denominator. For horizontal asymptotes, we will go according to the cases.

Complete step-by-step solution:
First, we will find the vertical asymptotes for rational functions. We will check whether there are any common factors in the numerator and denominator or not. If we find any common factors in the numerator and denominator, then we will cancel all the common factors.
Then, we will convert the denominator to zero. After that we will solve for \[x\]and will get vertical asymptotes for rational functions.
Now, we will find the horizontal asymptotes for rational functions. Now, we will try to make:
\[f(x) = \dfrac{{p(x)}}{{q(x)}}\]
Here, p(x) is the polynomial degree ‘m’ which is having ‘a’ as coefficient. The q(x) is the polynomial degree ‘n’ which is having ‘b’ as coefficient. Then we have three cases through which we will try to find the horizontal asymptotes for rational functions. The cases are:
First case: if m=n, then \[y = \dfrac{a}{b}\]. This is the horizontal asymptote of ‘f’.
Second case: if mThird case: if m
Note: The functions which are made by dividing two functions, those functions are called rational functions. We represent rational functions as \[dfrac{{f(x)}}{{g(x)}}\]. Here, f(x) and g(x), both are functions. All rational functions have asymptotes because when we plot the graph, we can see that they cannot touch the zones ever.