LIMITS AT INFINITY Consider the "end­behavior" of a function on an infinite interval.

In the example above, the value of y approaches 3 as x increases without bound. Similarly, f(x) approaches 3 as x decreases without bound.

NOTATION:

Means that the limit exists and the limit is equal to L.

1 HORIZONTAL ASYMPTOTE . . .

The height that a function tries to, but cannot, reach as the function's x­values get infinitely large or small.

Definition The line y = b is a horizontal asymptote of the graph of a function y = f (x) if either

1.

2.

EX.#1: A function can have more than one horizontal asymptote.

2 End Behavior Models

Ex #2: Find

Suppose x is large and positive, then . . .

and

So, for large positive values of x.

Suppose x is large and negative, then . . .

4x and 2x are still behavior models for the function, but the numerator and denominator will both be negative now.

So, for large positive values of x.

Then, while

Graph

Graph

EX 2 EX 2

3 Let's say we are trying to evaluate some function called r (x), which is defined as a fraction whose numerator, n (x), and denominator d (x) are simply polynomials. Compare the degrees (highest exponents) of n(x) and d (x).

r(x)= n(x) A picture for your d(x) head.

The guidelines below only apply to limits at infinity so be careful.

If degree of numerator is less than degree of denominator, then limit is zero. (BOBO Bigger On Bottom O)

If degree of numerator equals degree of denominator, then limit is the ratio of coefficients of the highest degree. (BETC Bottom Equals Top Coefficient)

If degree of numerator is greater than degree of denominator, then limit does not exist. (BOTU Bigger On Top Undefined).

Other functions, then consider size of the function: logarithmic < polynomial < exponential

4 EX #3: Evaluate the following limits:

5 Dividing Method for Finding Horizontal Asymptotes Remember ∞ is not a real number and should never be substituted for the variable x.

EX #4: Find

Graph

Graph

EX 4 EX 4

EX #5: Find

Graph

Graph

EX 5 EX 5

EX #6: Find

Graph

Graph

EX 6 EX 6

6 Summary:

Final Note: Be sure you see that the equal sign in the statement does not mean that the limit exists. On the contrary, it tells you how the limit fails to exist by denoting the unbounded behavior of f (x ) as x approaches c.

1. When the limit of a function at some x­value is approaching ±∞, then the function has a ______at that value.

2. To find a horizontal asymptote you should find the limit of f(x) as ______

3. A horizontal asymptote occurs at y = 0 when the degree of the ______is greater than the degree of the ______.

4. A limit does not exist when the degree of the ______is greater. Thus, no horizontal asymptote.

5. A horizontal asymptote occurs at the ______of leading coefficients when the highest degreed terms are ______.

7