Question

Explain the difference between finding the horizontal and vertical asymptotes.

When will there be a point of discontinuity?

What could the graph look like when there is no horizontal or vertical asymptotes.

Answer 1

The function has not been stated here, but we presume that the reference is to a rational function f(x) = p(x)/q(x), where all the 3 functions are functions of x.

1. When f(x) = p(x) / q(x), the equation(s) of the vertical asymptotes can be found by finding the roots of q(x), e.g. if x = a is a zero of q(x) , then x = a is a vertical asymptote of f(x).

The location of the horizontal asymptote is determined by looking at the degree(n) of the numerator p(x)   and the degree (m) of the denominator q(x).If n<m, the X-axis i.e. the line y = 0 is the horizontal asymptote.If n=m, then y=a_n / b_m is the horizontal asymptote i.e.the ratio of the leading coefficients. If n>m, there is no horizontal asymptote.

2. The zeros of the denominator q(x) are the points of discontinuity as division by 0 is not defined.

3. The graph of a rational function does not cross the asymptoes if there are any asymptotes. If there are no asymptotes, then the graph is not bound by any lines.