Section 2.5: Limits at Infinity
In 2.4, we looked at limits around vertical asymptotes – places
where the function value approaches infinity.
Now we’ll look at what happens as the independent variable (x,
for example) approaches infinity. This tells us about the “end
behavior” of a function.
What are the possibilities for the behavior of the output f(x) of
a function when x gets very large (or, what can possibly
happen to f(x) as x →∞)?
lim 𝑓 (𝑥 ) = ∞
𝑥→∞
lim 𝑓 (𝑥 ) does not exist
𝑥→∞
lim 𝑓 (𝑥 ) = 𝑐
𝑥→∞
(for some constant c)
lim 𝑓 (𝑥 ) = 𝑐
𝑥→∞
(for some constant c)
Note that in the fraction
𝑎
,
𝑏
with a, b ≠ 0 …
if the value of a stays mostly the same and b approaches infinity,
what happens to the value of
𝑎
?
𝑏
𝐄𝐱𝐚𝐦𝐩𝐥𝐞: Find lim 𝑓(𝑥 ).
𝑥→∞
7
(a) 𝑓(𝑥 ) = 5 + 3𝑥 2
(b) 𝑓(𝑥 ) = 5
7 −2
+ 3𝑥
𝐄𝐱𝐚𝐦𝐩𝐥𝐞: Find lim 𝑓(𝑥 ).
𝑥→∞
(c) 𝑓(𝑥 ) = 5
sin 𝑥 2
+ 3𝑥 2
𝐄𝐱𝐚𝐦𝐩𝐥𝐞: Find lim 𝑓(𝑥 ).
𝑥→∞
(d) 𝑓(𝑥 ) =
7𝑥 3 +2𝑥 2 −3𝑥+5
2𝑥 3 +𝑥
End Behavior of Polynomials
A polynomial of degree n:
As x →∞, or as x →-∞, polynomials will behave like anxn.
n even, an > 0
n even, an < 0
n odd, an > 0
n odd, an < 0
𝐄𝐱𝐚𝐦𝐩𝐥𝐞: Find lim 𝑓(𝑥 ) and lim 𝑓 (𝑥 ) .
𝑥→∞
𝑥→−∞
(a) 𝑓(𝑥 ) = 7𝑥 31 + 2𝑥 2 − 3𝑥 + 5
(b) 𝑓(𝑥 ) = −7𝑥 34 + 2𝑥 2 − 3𝑥 + 5
Rational Functions
We’ve seen:
7
𝐥𝐢𝐦
=0
𝒙→∞ 3𝑥 2
and
m<n
7𝑥 3 + 2𝑥 2 − 3𝑥 + 5 7
𝐥𝐢𝐦
=
3
𝒙→∞
2𝑥 + 𝑥
2
m=n
Hopefully you also believe this:
3𝑥 5 + 2𝑥
𝐥𝐢𝐦
=∞
2
𝒙→∞
7𝑥
m>n
(How can we verify this?)
There’s only one case we’re missing …
When m = n + 1, the rational function has a slant asymptote … but
that does not affect the limit as x approaches infinity.
Be sure to look over Theorem 2.8 in your text: End
behavior of e x, e -x, and ln x.
ex
e –x
ln x