1. Definition of Limit
Lesson 2-2

2. Verbal Definition of Limit
L is the limit of f (x) as x approaches c
if and only if
L is the one number you can keep f (x)
arbitrarily close to just by keeping x close to c, but not equal to c.

3. Example: A rational algebraic function
Graph it!
Yes! The denominator would go to zero. Then you would be dividing by zero. Yuck!!!!
Hint
Trace to x = -2
Why is there no value for the function at x = -2?

4. Explore f (x) at x = -2 Algebraically, Indeterminate Form
Looking back at the graph, what number does f (x) appear to be at x = -2?

5. Explore f (x) at x = -2 Tabular
y 1
-2.03
1.94
-2.02
1.96
-2.01
1.98
-2.
undef
-1.99
2.02
In TBLSET set tblStart… Δ……..0.01
Press ENTER. Then view the TABLE.
Tabular
Looking at the table, what number does f (x) appear to be at x = -2?
Approaches 2

6. How about finding the limit algebraically?
Looking at both graph and table of f(x) we can conclude the limit as x approaches -2 is 2. x
y 1
-2.03
1.94
-2.02
1.96
-2.01
1.98
-2.
undef
-1.99
2.02
How about finding the limit algebraically?

7. Finding a limit algebraically…
Remember
For the original equation,
x at -2 does not exist. We say its approaching a value of 2.

8. Removable Discontinuity
The function is discontinuous because of the gap at x = -2 but the gap can be removed by defining f (-2) to be 2.
An open circle at the point of discontinuity is used to illustrate that.

9. Step Discontinuity This function is discontinuous
because of the gap at x = 0.
However, it can not simply be removed since there is a large “step” between the two branches.
What is the limit at x = 2?
What is the limit at x = 0?
Answer: -1
The discontinuity is at x = 0. As x approaches 2 from the left, g(x) is close to -1. As x approaches 2 from the right, g(x) is still close to -1.
Answer:
Does not exist.

10. Formal Definition of Limit:
L is the limit of f (x) as x approaches c
if and only if
for any positive number epsilon (ε),
no matter how small, there is a positive
number delta (δ)
such that
if x is within delta units of c (but not equal to c)
then f (x) is within epsilon units of L.

11. Formal Definition of Limit
L is the limit of f (x) as x approaches c
if and only if
for any positive number ε,
no matter how small,
L is 4
c is 2
ε is 1 “arbitrarily”
δ is 0.8 and 0.6
there is a positive number δ such that
if x is within δ units of c (but not equal to c)
then f (x) is within ε units of L
Pick the δ that is most restrictive.
So δ is 0.6

12. Infinite Discontinuity
What is the limit at x = 2?
Infinite Discontinuity
Answer: No limit
Removable
Discontinuity
What is the limit at x = 5?
Answer: 2 The value of h(5) is 4 but h(x) approaches 2 from the left and from the right.

13. Example: Given Answer: 7 0/0 called Indeterminate Form
From the graph, what do you think the limit of f (x) is as x approaches 1?
Answer: 7
0/0 called
Indeterminate Form
Try to evaluate f (1) by direct substitution. What form does the answer take? What is this form called?

14. Limit is 7 as x approaches 1
Factor the numerator and simplify the expression. Although the simplified expression does not equal f (1), you can substitute 1 for x and get an answer. What is the answer and what does it represent?
Limit is 7
as x approaches 1 Ω