How to find the vertical and horizontal asymptotes of a rational function? 

We need to make sure to check for all the removable discontinuities from the rational function.

Answer: We can find the asymptotes by calculating the zeros of the denominator and checking for the degrees of numerator and denominator.

Both the vertical and horizontal asymptotes describe the nature of the graph as the input gets very large or very small.

Explanation:

A vertical asymptote to a rational function may be found by taking a quick view of the factors of the denominator that are similar to that of the factors present in the numerator. Vertical asymptotes occur at zeros of such factors.

Steps to encounter the vertical asymptotes in a function include :

Firstly, factor both the numerator and denominator.

Check, if any, restrictions in the domain of the function.

Reduce the expression by canceling the common factors.

Note the values at which the denominator occurs to zero in the simplified version. These are where the vertical asymptotes occur.

For checking horizontal asymptotes, there are three different outcomes, we need to check.

Case 1) If the degree of the denominator > degree of the numerator there is a horizontal asymptote at y = 0.

Case 2) If the degree of the denominator < degree of the numerator by one, we get a slant asymptote.

Case 3) If the degree of the denominator = degree of the numerator, there is a horizontal asymptote at y = a_n / b_n , where a_n, b_n are the leading coefficients of p (x) and q (x) for f (x) = p (x) / q (x) , q (x) ≠ 0.

Hence, the above-mentioned cases for both horizontal and vertical asymptotes clearly show the path to search for them in a rational function.