http://www.ck12.org Chapter 1. Functions, Limits, and Continuity
Since our original function was roughly of the form f(x) =^1 x, this enables us to determine limits for all other
functions of the formf(x) =x^1 pwithp> 0 .Specifically, we are able to conclude that limx→∞x^1 p=0. This
shows how we can find infinite limits of functions by examining the 1 end behaviorof the function f(x) =
xp,p>^0.
The following example shows how we can use this fact in evaluating limits of rational functions.Example 1:
Find limx→∞x (^6) −^2 xx^35 −+x 32 x+ (^4) −x− 2 x^1 + 1.
Solution:
Note that we have the indeterminate form, so Limit Property #5 does not hold. However, if we first divide both
numerator and denominator by the quantityx^6 , we will then have a function of the form
f(x)
g(x)=
(^2) xx 63 −xx 62 +xx 6 −x (^16)
xx^66 −xx 65 + (^3) xx 64 − (^2) xx 6 +x 16 =
x^23 −x^14 +x^15 −x^16
1 −^1 x+x^32 −x^25 +x^16.
We observe that the limits limx→∞f(x)and limx→∞g(x)both exist. In particular, limx→∞f(x) =0 and limx→∞g(x) =
Hence Property #5 now applies and we have limx→∞x (^6) −^2 xx^35 −+x 32 x+ (^4) −x− 2 x^1 + 1 =^01 =0.
Lesson Summary
We learned to find infinite limits of functions.
We analyzed properties of infinite limits.
We identified asymptotes of functions.
We analyzed end behavior of functions.
Multimedia Links
For more examples of limits at infinity(1.0), see Math Video Tutorials by James Sousa, Limits at Infinity (9:42).
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/532Review Questions
In problems 1 - 7, find the limits if they exist.
- limx→ 3 +(x(−x+ 2 )^22 )−^21