1.2 Finding Limits Numerically and Graphically.

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Presentation transcript:

1.2 Finding Limits Numerically and Graphically

Limits A function f(x) has a limit L as x approaches c if we can get f(x) as close to c as possible but not equal to c. x is very close to, not necessarily at, a certain number c NOTATION:

3 Ways to find Limits Numerically - construct a table of values and move arbitrarily close to c Graphically - exam the behavior of graph close to the c Analytically

1) Given , find 2 x 1.9 1.99 1.999 1.9999 3.61 3.9601 3.996001 3.99960001 4 2 x 2.0001 2.001 2.01 2.1 4.004001 4.0401 4.41 4.00040001 4

2) Given , find 1 x 0.9 0.99 0.999 0.9999 2.710 2.9701 2.997001 2.99970001 3 1 x 1.0001 1.001 1.01 1.1 3.003001 3.0301 3.31 3.00030001 3

3. What does the following table suggest about x 0.9 0.99 0.999 1.001 1.01 1.1 F(x) 7 25 4317 3.0001 3.0047 3.01

Finding Limits Graphically There is a hole in the graph. Limits that Exist even though the function fails to Exist

One sided Limits notation Limits from the right Limits from the left

4) Use the graph of to find

5) Use the graph of to find

6) Finding Limits graphically a) Find b) Find c) Find d) Find e) Find

7) Use the graph of to find 1 –1 Does Not Exist – DNE

Limits that Fail to Exist In order for a limit to exist the limit must be the same from both the left and right sides. 1 –1

Limits that Fail to Exist The behavior is unbounded or approaches an asymptote 1 –1

Limits that Fail to Exist The behavior oscillates

HOMEWORK Page 54 # 1-10 all numerically # 11 – 26 all graphically