Section 3.3: Limits at Infinity
In many biological problems, we are interested in the long-term behavior of a function.
For instance, will a population persist or become extinct in the long-run? What is the
concentration of medicine in a patient’s bloodstream in the long-run?
Definition: If the values of f (x) can be made arbitrarily close to L by taking x sufficiently
large, then we say that the limit of f (x), as x approaches ∞, equals L. This is denoted by
lim f (x) = L.
x→∞
Similarly, if the values of f (x) can be made arbitrarily close to L by taking x sufficiently
large negative, then we say that the limit of f (x), as x approaches negative −∞, equals L.
This is denoted by
lim f (x) = L.
x→−∞
Example: Evaluate the following limits.
1
x→∞ x
(a) lim
(b) lim
x→−∞
1
x3
1
7x3 + 4x
x→∞ 2x3 − x2 + 3
(c) lim
3 − x2
x→−∞ 1 − 2x2
(d) lim
2x2 + 5x − 6
x→∞ 4x3 − 2x2 + 1
(e) lim
x3 − 2x + 3
x→∞
4 − 3x2
(f) lim
2
Theorem: (Limit of an Exponential Function at Infinity)
lim ex = ∞
x→∞
and
lim e−x = 0.
x→∞
Example: Evaluate the following limits, if they exist.
e−x
x→∞ 1 − e−x
(a) lim
ex
x→∞ 2 − ex
(b) lim
(c) lim
x→−∞
4
1 + e−x
3
Example: (Logistic Growth) Suppose that the size of a population at time t is given by
N (t) =
100
t ≥ 0.
1 + 9e−t
Sketch the graph of N (t) and determine the limiting population size.
4