End Behavior
The behavior of a function as x→±∞x→±∞ is called the function’s end behavior. At each of the function’s ends, the function could exhibit one of the following types of behavior:
- The function f(x)f(x) approaches a horizontal asymptote y=Ly=L.
- The function f(x)→∞f(x)→∞ or f(x)→−∞f(x)→−∞.
- The function does not approach a finite limit, nor does it approach ∞∞ or −∞−∞. In this case, the function may have some oscillatory behavior.
Let’s consider several classes of functions here and look at the different types of end behaviors for these functions.
End Behavior for Polynomial Functions
Consider the power function f(x)=xnf(x)=xn where nn is a positive integer. From Figure 11 and Figure 12, we see that,
and,


Using these facts, it is not difficult to evaluate limx→∞cxnlimx→∞cxn and limx→−∞cxnlimx→−∞cxn, where cc is any constant and nn is a positive integer.
evaluating limits of power functions
If c>0c>0, the graph of y=cxny=cxn is a vertical stretch or compression of y=xny=xn, and therefore,
If c<0c<0, the graph of y=cxny=cxn is a vertical stretch or compression combined with a reflection about the xx-axis, and therefore,
If c=0,y=cxn=0c=0,y=cxn=0, in which case limx→∞cxn=0=limx→−∞cxnlimx→∞cxn=0=limx→−∞cxn.
For each function ff, evaluate limx→∞f(x)limx→∞f(x) and limx→−∞f(x)limx→−∞f(x).
- f(x)=−5x3f(x)=−5x3
- f(x)=2x4f(x)=2x4
We now look at how the limits at infinity for power functions can be used to determine limx→±∞f(x)limx→±∞f(x) for any polynomial function ff.
Consider a polynomial function
of degree n≥1n≥1 so that an≠0. Factoring, we see that,
As x→±∞, all the terms inside the parentheses approach zero except the first term. We conclude that,
The function f(x)=5x3−3x2+4 behaves like g(x)=5x3 as x→±∞ as shown below.

x | 10 | 100 | 1000 |
f(x)=5x3−3x2+4 | 4704 | 4,970,004 | 4,997,000,004 |
g(x)=5x3 | 5000 | 5,000,000 | 5,000,000,000 |
x | −10 | −100 | −1000 |
f(x)=5x3−3x2+4 | −5296 | −5,029,996 | −5,002,999,996 |
g(x)=5x3 | −5000 | −5,000,000 | −5,000,000,000 |