Limits at Infinity and Asymptotes: Learn It 3

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:

  1. The function f(x)f(x) approaches a horizontal asymptote y=Ly=L.
  2. The function f(x)f(x) or f(x)f(x).
  3. 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,

limxxn=;n=1,2,3,limxxn=;n=1,2,3,

and,

limxxn={;n=2,4,6,;n=1,3,5,limxxn={;n=2,4,6,;n=1,3,5,
The functions x2, x4, and x6 are graphed, and it is apparent that as the exponent grows the functions increase more quickly.
Figure 11. For power functions with an even power of nn, limxxn==limxxnlimxxn==limxxn.
The functions x, x3, and x5 are graphed, and it is apparent that as the exponent grows the functions increase more quickly.
Figure 12. For power functions with an odd power of nn, limxxn=limxxn= and limxxn=limxxn=.

Using these facts, it is not difficult to evaluate limxcxnlimxcxn and limxcxnlimxcxn, 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,

limxcxn=limxxnlimxcxn=limxxn and limxcxn=limxxnlimxcxn=limxxn if c>0c>0

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,

limxcxn=limxxnlimxcxn=limxxn and limxcxn=limxxnlimxcxn=limxxn if c<0c<0

If c=0,y=cxn=0c=0,y=cxn=0, in which case limxcxn=0=limxcxnlimxcxn=0=limxcxn.

For each function ff, evaluate limxf(x)limxf(x) and limxf(x)limxf(x).

  1. f(x)=5x3f(x)=5x3
  2. 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

f(x)=anxn+an1xn1++a1x+a0f(x)=anxn+an1xn1++a1x+a0
 

of degree n1n1 so that an0. Factoring, we see that,

f(x)=anxn(1+an1an1x++a1an1xn1+a0an1xn).
 

As x±, all the terms inside the parentheses approach zero except the first term. We conclude that,

limx±f(x)=limx±anxn.

The function f(x)=5x33x2+4 behaves like g(x)=5x3 as x± as shown below.

Both functions f(x) = 5x3 – 3x2 + 4 and g(x) = 5x3 are plotted. Their behavior for large positive and large negative numbers converges.
Figure 13. The end behavior of a polynomial is determined by the behavior of the term with the largest exponent.
Table 1. A polynomial’s end behavior is determined by the term with the largest exponent.
x 10 100 1000
f(x)=5x33x2+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)=5x33x2+4 5296 5,029,996 5,002,999,996
g(x)=5x3 5000 5,000,000 5,000,000,000