1. We have an excluded value/point discontinuity at x = 1.
What happens as
from the left and from the right?
as x approaches 1 x
f(x) ?
f(x)
aproaches 3

2. If a function has no discontinuity (excluded value), then we can just plug in the x value we are approaching:
Ex. Find the limit.

3. Ex. Evaluate the function
at several points near
x = 0 and use the results
to find the limit. x
1.9949
1.9995 ?
f(x)
2.0049
f(x) approached 2
f(x) approached 2

4. Ex. Evaluate the function
at several points near
x = 0 and use the results
to find the limit. x ?
f(x)

5. Ex. Find the limit as x 2 where f(x) =
What is the y-value as x approaches 2 from the
left and from the right?
The limit is 1 since f(x) = 1 from the left and from
the right as x approaches 2.
The value of f(2) is immaterial!!!

6. LIMTS THAT DO NOT EXIST: 3 types of limits that fail to exist.
Behavior that differs from the left and from
the right.
Ex.
the limit D.N.E. , since the limit from the left does
not = the limit from the right.

7. 2. Unbounded behavior
Ex.
Since f(x)
the limit D.N.E.

8. 3. Oscillating behavior (use calculator)
As x , f(x) oscillates between –1 and 1, therefore
the limit D.N.E.
Limits D.N.E. when:
f(x) approaches a different number from the
right side of c than it approaches from the left side.
2. f(x) increases or decreases without bound as x
approaches c.
f(x) oscillates between two fixed values as x
approaches c.

9. One-Sided Limits
We can say that a limit exists from one side or the other as long as we specify which direction we are coming from
Ex. -1 1
the left-hand and the right-hand limits do exist