Section 2.5

Limits at Infinity

Understanding horizontal asymptotes and the long-term behavior of functions as .

Rational Functions and Dominance

For rational functions, the limit at infinity is determined by the "struggle for dominance" between the numerator and denominator.

Top Heavy

Degree N > D

Bottom Heavy

Degree D > N

Balanced

Degree N = D

Methodology

Divide every term by the highest power of in the denominator. This uses the axiom that .

Beyond polynomials, students must understand the hierarchy of function growth:

An exponential function will eventually dominate any polynomial, no matter how large the degree.

Polynomial Time

Considered "efficient" — algorithms finish in reasonable time.

Exponential Time

Considered "intractable" — runtime explodes for large inputs.

The limit formalizes saying an algorithm "runs in quadratic time."

In biology, exponential growth is unrealistic over long periods. The Logistic Growth Model introduces a limiting factor:

Analysis

As , we have (assuming decay).

Therefore:

Interpretation: is the Carrying Capacity—the maximum population the environment can sustain.

The number itself is defined by a limit at infinity, originating from Jacob Bernoulli's study of compound interest: