2.4 Limits at Infinity

Summary

The notion of limit is extended to include points x=a where the limit does not exist but where the notation for plus or minus infinity can be used to provide additional information about the behavior of the function. The definition of limit is also extended to include limits as xgets large or small without bound.

By the end of your studying, you should know:

How to identify a case where we write that the limit equals plus infinity or minus infinity.
That if we write that the limit equals plus or minus infinity, the limit does not exist.
How to evaluate limits of quotients of polynomials.
How to evaluate limits as x approaches plus or minus infinity.
How to find horizontal and vertical asymptotes.

On-screen applet instructions: The slider generates values of x linearly from x = 0 to x = 20. At this value of x the slider is 3/4 of the way to the right. After that the values of x increase exponentially as the slider is moved the remaining distance to the right end. This allows us to explore "large" values of x.

Examples

Find the horizontal and vertical asymptotes of

Allyson carries an 80 degree cup of coffee into a room which has been heated to 20 degrees. According to Newton's Law of Cooling, the temperature of the coffee at time t will be

where t is the amount of time the cup of coffee is sitting out in the room. Find

the temperature the coffee will reach if it is left in the room indefinitely.

Population growth is often modeled by the equation

where C is the carrying capacity of the population, and k and A are constants controlling details about the population. For example, in an environment which has a carrying capacity of 1000 fish, the population equation might be

Find