Find the end behavior of a rational function
A curve far to the right
The curve shows \(f(x) = \dfrac{4x^2 + 3x}{2x^2 + 7}\) for x from 0 to 100. The top and the bottom both grow as x grows, so it is not clear what the fraction does.
Drag the orange point to the right, or use the slider. The readouts give the top, the bottom and the height of the curve at the point.
The height is about 2.077 at \(x = 10\), 2.027 at \(x = 50\) and 2.014 at \(x = 100\), so it comes closer to 2. The toggle marks the height 2 with a dashed line.
When x is large, \(3x\) is small next to \(4x^2\): at \(x = 100\), it is 300 against 40,000. In the same way, 7 is small next to \(2x^2\), so the fraction acts like \(4x^2/2x^2 = 2\).
Any fraction of polynomials
The curve is the graph of a rational function, which is a fraction of two polynomials. The leading term of a polynomial has the highest power of x, and that power is its degree. Here the leading terms are \(a x^m\) on the top and \(b x^n\) on the bottom.
Each button sets the degree of the bottom lower than, equal to, or higher than the degree of the top.
If the degrees are equal, the curve approaches the height \(a/b\). The numbers a and b are the leading coefficients, which multiply the highest powers.
If the bottom has the higher degree, the curve approaches the x-axis, which turns orange. If the bottom has the lower degree, the curve climbs without bound.
Far to the right, the other terms are small next to the leading terms, so only the leading terms decide the curve.
Notation
To ask which height the curve approaches, write a limit. The symbol \(x \to \infty\) means that x is positive and grows without bound.
\[\lim_{x \to \infty} \frac{4x^2 + 3x}{2x^2 + 7}\]Divide the top and the bottom by the highest power of x in the bottom, here \(x^2\). The fraction keeps its value, because both parts are divided by the same number.
\[= \lim_{x \to \infty} \frac{4 + \cq{3/x}}{2 + \cq{7/x^2}}\]A fixed number divided by a very large number is small, so \(3/x\) and \(7/x^2\) approach 0.
\[= \frac{4 + 0}{2 + 0} = \cy{2}\]If the degrees are equal, the limit is the ratio of the leading coefficients. If the bottom has the higher degree, the limit is 0. If the top has the higher degree, there is no finite limit.
\[\frac{a x^m + \cdots}{b x^n + \cdots} \to \begin{cases} a/b & m = n \\ 0 & m < n \\ \text{infinite} & m > n \end{cases}\]
When it fails
If you divide the top and the bottom of \(f(x) = \dfrac{3x^2 + 1}{x + 2}\) by x, the result is \(\dfrac{3x + 1/x}{1 + 2/x}\). The top grows without bound and the bottom approaches 1.
The curve climbs out of the top of the figure and reaches about 294 at x = 100, so there is no finite limit.
Example 1: equal degrees
Find \(\displaystyle\lim_{x \to \infty} \frac{8x - 3}{2x + 5}\).
The top and the bottom both have degree 1. Divide both by x, the highest power in the bottom.
\[\frac{8x - 3}{2x + 5} = \frac{8 - \cq{3/x}}{2 + \cq{5/x}}\]As x grows without bound, \(3/x\) and \(5/x\) approach 0.
\[\to \frac{8 - 0}{2 + 0}\]The limit is 4, the ratio of the leading coefficients 8 and 2. The curve approaches the dashed orange line.
\[\lim_{x \to \infty} \frac{8x - 3}{2x + 5} = \cy{4}\]
Example 2: a higher degree in the bottom
Find \(\displaystyle\lim_{x \to \infty} \frac{x^2 + 2}{5x^3 - x}\).
The top has degree 2 and the bottom has degree 3. Divide both by \(x^3\), the highest power in the bottom.
\[\frac{x^2 + 2}{5x^3 - x} = \frac{\cq{1/x} + \cq{2/x^3}}{5 - \cq{1/x^2}}\]A tempting wrong move: The leading coefficients are 1 and 5, so the limit is 1/5.
\[\lim = \cr{1/5}\]The curve does not approach 0.2. At \(x = 100\) its height is about 0.002, far below the red line, and it continues to fall.
The ratio of the leading coefficients applies only when the degrees are equal. Here the top approaches 0 and the bottom approaches 5, so the limit is 0/5 = 0.
\[\lim_{x \to \infty} \frac{x^2 + 2}{5x^3 - x} = \cy{0}\]
Practice
Find \(\displaystyle\lim_{x \to \infty} \frac{12x^2 - 5x + 1}{4x^2 + 9}\).
Compare the degrees first. Move the point to the right to see the height that the curve approaches.
Solution
Both degrees are 2. Divide the top and the bottom by \(x^2\).
\[\frac{12 - \cq{5/x} + \cq{1/x^2}}{4 + \cq{9/x^2}}\]The terms \(5/x\), \(1/x^2\) and \(9/x^2\) approach 0, so the limit is the ratio of the leading coefficients 12 and 4.
\[\frac{12}{4} = \cy{3}\]
Questions
The graph shows \(h(x) = \dfrac{6x^2 + 5}{3x^2 + 4}\) for x from \(-10\) to 10.
As x grows to the right, which height does the curve approach?
- 2
- 6
- 1.25
- 0.5
Answer
2
As x grows to the left, which height does the curve approach?
- 2
- −2
- 0
- There is no limit
Answer
2
What is \(\displaystyle\lim_{x \to \infty} \frac{4x + 1}{x^2 - 6}\)?
- 0
- 4
- −1/6
- The limit is infinite
Answer
0
Summary
You divide the top and the bottom by the highest power of x in the bottom.
This works because the lower powers are small next to the leading terms when x is large.
This gives the height that a rational function approaches far to the right, such as y = 2 here.
This is one of four skills in topic 1.15 of Which & Why: AP Calculus.