Section 1 2 finding limits graphically and numerically
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Section 1.2 - Finding Limits Graphically and Numerically. 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. f ( x ). L. x. c. The limit of f(x)….

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Limit
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.

f(x)

L

x

c

The limit of f(x)…

is L.

Notation:

as x approaches c…


Calculating limits
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


Example 1
Example 1

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

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.


Example 2
Example 2

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

Can’t divide by 0

-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.


Example 3
Example 3

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

-6

2.9

2.99

-6

2.999

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 -4 changes.


Three limits that fail to exist
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.


Three limits that fail to exist1
Three Limits that Fail to Exist

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


Three limits that fail to exist2
Three Limits that Fail to Exist

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

Closest

Closer

Close


A limit that does exist
A Limit that DOES Exist

If the domain is restricted (not infinite), the limit off(x)exists as x approaches an endpoint of the domain.


Example 11
Example 1

Given the function t defined by the graph, find the limits at right.


Example 21
Example 2

Sketch a graph of the function with the following characteristics:

  • does not exist, Domain: [-2,3), and Range: (1,5)

  • does not exist, Domain: (-∞,-4)U(-4,∞), and Range: (-∞,∞)


Classwork
Classwork

Sketch a graph and complete the table to find the limit (if it exists).

2.8680

2.732

2.7196

DNE

2.7169

2.7048

2.5937

This a very important value that we will investigate more in Chapter 5. It deals with natural logs.

Why is there a lot of “noise” over here?


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