How do I find the horizontal limits of f(x)=(2x^3-11x^2-11x)/(5-9x-6x^3)?

Question

I'm not entirely sure what it is wanting me to do. There doesn't seem to be much to factor other than the x out of the top part. What exactly do I do to find horizontal limits?

Scarlett Allison · Accepted Answer

See below.

The question is really asking you to find the limit to infinity. This is sometimes called the end behaviour of a polynomial. When we have rational functions like the one given, the limit to infinity will tend towards a horizontal line, unless the degree of the numerator is higher than the degree of the denominator, in which case it tends towards an oblique asymptote. To find this, we only need to concentrate on the terms of the highest degree in the numerator and denominator.

These terms increase more rapidly than the terms of lower degree.

From the given function we have:

#(2x^3)/(-6x^3)#

This cancels to:

#-1/3#

So the limit to infinity is:

#lim_(x->+-oo)(2x^3-11x^2-11x)/(5-9x-6x^3)=-1/3#

This means as #x->+-oo# #\ \ \ \ \ \ \ f(x)->-1/3#

This is the horizontal asymptote:

#y=-1/3#

The graph confirms this:

Adam Adkins · Answer

undefined To find the horizontal limits of the function $ f(x) = \frac{2x^3 - 11x^2 - 11x}{5 - 9x - 6x^3} $, you need to determine what happens to the function as $ x $ approaches positive or negative infinity.

To find the horizontal limits:
1. Check the degree of the numerator and the denominator. If the degree of the numerator is greater than or equal to the degree of the denominator, the horizontal limit is either positive or negative infinity.
2. If the degree of the numerator is less than the degree of the denominator, the horizontal limit is zero.

In this case:
- The degree of the numerator is 3.
- The degree of the denominator is 3.

Since the degrees are equal, we need to perform the division of the leading coefficients to find the horizontal limit.

After simplifying the expression, if the horizontal limit exists, it will be the ratio of the leading coefficients after simplification.

So, divide the leading coefficient of the numerator by the leading coefficient of the denominator to find the horizontal limit.