1. Section 1.2 – Finding Limits Graphically and Numerically

2. Informal Definition of a Limit
If f(x) becomes arbitrarily close to a single REAL number L as x approaches c from either side, the limit of f(x), as x appraches c, is L. c L
f(x) x
The limit of f(x)…
is L.
Notation:
as x approaches c…

3. Calculating Limits Our book focuses on three ways:
Graphical Approach – Draw a graph
Numerical Approach – Construct a table of values
Analytic Approach – Use Algebra or Calculus
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4. Example
Given the function t defined by the graph, find the limits at right.

5. Example 1
Use the graph and complete the table to find the limit (if it exists). x
1.9
1.99
1.999 2
2.001
2.01
2.1
f(x)
6.859
7.88
7.988 8
8.012
8.12
9.261
If the function is continuous at the value of x, the limit is easy to calculate with direct substitution: 23 = 8.

6. Example 2
Use the graph and complete the table to find the limit (if it exists).
Can’t divide by 0 x
-1.1
-1.01
-1.001 -1
-.999
-.99
-.9
f(x)
-2.1
-2.01
-2.001
DNE
-1.999
-1.99
-1.9
If the function is not continuous at the value of x, a graph and table can be very useful.

7. The limit does not change if the value at x=-4 changes.
Example 3
Use the graph and complete the table to find the limit (if it exists). -6 x
-4.1
-4.01
-4.001 -4
-3.999
-3.99
-3.9
f(x)
2.9
2.99
2.999 -6 8
2.999
2.99
2.9
If the function is not continuous at the value of x, the important thing is what the output gets closer to as x approaches the value.
The limit does not change if the value at x=-4 changes.

8. Three Limits that Fail to Exist
f(x) approaches a different number from the right side of c than it approaches from the left side.

9. Three Limits that Fail to Exist
f(x) increases or decreases without bound as x approaches c.

10. Three Limits that Fail to Exist
f(x) oscillates between two fixed values as x approaches c.
Closest
Closer
Close x
f(x) -1 1
DNE