Section 1.2 - Finding Limits Graphically and Numerically

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Section 1.2 - Finding Limits Graphically and
Numerically
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Limit

• Informal Definition 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.

The limit of f(x)
is L.
Notation
as x approaches c
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Calculating Limits

• Our book focuses on three ways
• Numerical Approach Construct a table of values
• Graphical Approach Draw a graph
• Analytic Approach Use Algebra or calculus

This Lesson
Next Lesson
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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
9.261
8.12
8.012
If the function is continuous at the value of x,
the limit is easy to calculate.
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Example 2

• Use the graph and complete the table to find the
  limit (if it exists).

Cant divide by 0
x -1.1 -1.01 -1.001 -1 -.999 -.99 -.9
f(x)
-2.1
-2.01
-2.001
DNE
-1.9
-1.99
-1.999
If the function is not continuous at the value of
x, a graph and table can be very useful.
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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)
-6
2.9
2.99
2.999
8
2.9
2.99
2.999
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 -4
changes.
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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.

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Three Limits that Fail to Exist

• f(x) increases or decreases without bound as x
  approaches c.

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Three Limits that Fail to Exist

• f(x) oscillates between two fixed values as x
  approaches c.

Closest
Closer
Close
x 0
f(x) -1 1 -1 DNE 1 -1 1
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A Limit that DOES Exist
If the domain is restricted (not infinite), the
limit of f(x) exists as x approaches an endpoint
of the domain.
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Example 1
Given the function t defined by the graph, find
the limits at right.