Euclid was a Greek mathematician best known for his treatise on geometry: The Elements. This influenced the development of Western mathematics for more than 2000 years. Euclid was a Greek mathematician best known for his treatise on geometry: The Elements. This influenced the development of Western mathematics for more than 2000 years. Euclid of Alexandria 325 – 265 B.C. Euclid of Alexandria 325 – 265 B.C.

So we say, This limit shows that f (x) has a horizontal asymptote at y = 1.

We take limits to infinity to analyze end-behavior of functions.

Left-Hand Behavior Right-Hand Behavior

Example 1

Example 2

As the denominator gets larger, the value of the fraction gets closer to zero. So by definition, y = 0 is a horizontal asymptote. Recall vertical asymptote at x = 0. horizontal asymptote at y = 0. Example 3

Find: x y When we graph this function, the limit appears to be zero. Example 4 (2)

Thus, by the Squeeze Theorem: x y Example 4 When we graph this function, the limit appears to be zero. Find:

Example 5 Evaluate the following limit

Example 6 Dominating term

Method IMethod II Algebraic Limits

Dominating term effect (DTE) we get, dominant terms in numerator and denominator Example 7

Recall: Example

Example 8

Example 9 Hence, y = 0 is a horizontal asymptote. a.

Example 9 There is no left hand behavior because of the domain restriction. So, x = 0 is a vertical asymptote. Hence, there is no horizontal asymptote.

Example 9

Example 11 Example 10

a. Find the domain of f (x). b. Determine any asymptotes for f (x). Example 12 Example 13 Determine if a steady state exists and give the steady-state value for the function below.