Question

# What is the horizontal asymptote of the rational function

## The correct answer is: y = a / d

### A rational function is a function that is the ratio of polynomials. Any function of one variable, x, is called a rational function if, it can be represented as f(x) =, where p(x) and q(x) are polynomials such that q(x) ≠ 0.

Rational functions are of the form y = f(x)y = fx , where f(x)fx is a rational expression .

- If both the polynomials have the same degree, divide the coefficients of the leading terms. This is your asymptote.
- If the degree of the numerator is less than the denominator, then the asymptote is located at y = 0 (which is the x-axis).
- If the degree of the numerator is greater than the denominator, then there is no horizontal asymptote.

For the given function both the polynomials have the same degree, divide the coefficients of the leading terms.

y = a / d

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Let's say that r is a rational function.

Identify R's domain.

If necessary, reduce r(x) to its simplest form.

Find the x- and y-intercepts of the y=r(x) graph if one exists.

If the graph contains any vertical asymptotes or holes, locate where they are.

Then, identify and, if necessary, analyze r's behavior on each side of the vertical asymptotes.

Investigate R's final behavior. If one exists, locate the horizontal or slant asymptote.

The graph of y=r(x) can be drawn using a sign diagram and additional points if necessary.

### When will the graph of a rational function have no vertical asymptotes ? Give an example of such a function.

Let's say that r is a rational function.

Identify R's domain.

If necessary, reduce r(x) to its simplest form.

Find the x- and y-intercepts of the y=r(x) graph if one exists.

If the graph contains any vertical asymptotes or holes, locate where they are.

Then, identify and, if necessary, analyze r's behavior on each side of the vertical asymptotes.

Investigate R's final behavior. If one exists, locate the horizontal or slant asymptote.

The graph of y=r(x) can be drawn using a sign diagram and additional points if necessary.