1. Evaluating Limits Numerically & Intro into Algebraic
TS: Explicitly assessing information and drawing conclusions
Objective: To be able to evaluate a limit graphically and analytically.

2. General Limit
A general limit exists on f (x) when x = c, if the left- and right-hand limits are both equal there.
Mathematic Notation:
In other words:
f (x)  L as x  c

3. Limits Graphically 6 6 6 6

4. Limits Graphically
Undefined 2 2 2

5. Limits Graphically 5 8 8 8

6. Limits Graphically
– 3 3 7
DNE

7. Possible ways to evaluate a limit without a graph.
Substitution
Factoring
Conjugate
Table

8. value is the limit of the
Finding Limits
= 7
= 7
= 7
If a function
approaches the same
value from both
directions, then that
value is the limit of the
function at that point. x y x y .9
6.71
1.1
7.31
.99
6.9701
1.01
7.0301
.999
6.997
1.001
7.003

9. Finding Limits = DNE or NL = 3 = –3 If the Left-hand limit and the
Right-hand limit
are not equal,
the general limit
does not exist. x y x y
–1.1
3.1
–.9
–2.9
–1.01
3.01
–.99
–2.99
–1.001
3.001
–.999
–2.999

10. Finding Limits = DNE or NL = -∞ = ∞ x y x y If either the
Left-hand limit,
Right-hand limit, or
both do not exist,
the general limit
will not exist.
2.9
–44.1
3.1
56.1
2.99
–494
3.01
506.01
2.999
–4994
3.001
5006

11. How should we best use our calculator to help us if we need to make a table?