Visualization and interpretation of the limit

The limit has a finite value L The limit is at infinity

In the next slide we present the visual representation of this limit the values of f approach L x increases without bound In the next slide we present the visual representation of this limit

f (x) x x x L from the right as

This limit is telling us that the function f (x) has a horizontal asymptote y = L in the positive direction We show in the next slides other sketches of possible functions whose limit at infinity is L.

L f (x) x x x from the left as

f (x) L f (x) from right and left as

It is not true that when a function has a horizontal asymptote its graph does not touch the horizontal asymptote for large values of x, they may or may not. We give next some examples of different interpretations of this limit

If T is the temperature in degrees centigrade and t is time in minutes, then the interpretation of the limit is The temperature stabilizes around L degrees centigrade After a long time

If P is the price of a commodity in dollars and s is the supply, then the interpretation of the limit is its price stabilizes around 0 dollars As supply of the commodity increases without bound If the commodity is abundant then it becomes worthless