Finding Limits Graphically and Numerically
2015 Limits Introduction
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1.2
Estimate a limit using a numerical or graphical approach.
Learn different ways that a limit can fail to exist.
Study and use the informal definition of limit.
Objectives
Formal definition of a Limit:
If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit of f(x), as x approaches c, is L.
“The limit of f of x as x approaches c is L.”
limx cf x L
This limit is written as
Limits can be found in various ways:
a) Graphically
b) Numerically
c) Algebraically
Ex: Find the following limit:
2
2lim 1xx
5
0
sinlimx
x
x
An Introduction to Limits
Ex: Find the following limit:
sin /y x x
Looks like y=1
0
sinlim 1x
x
x
3
1
1lim
1x
x
x
An Introduction to Limits
Ex: Find the following limit:
Start by sketching a graph of the function
For all values other than x = 1, you can use standard
curve-sketching techniques.
However, at x = 1, it is not clear what to expect.
We can find this limit numerically:
An Introduction to Limits
To get an idea of the behavior of the graph of f near x = 1, you can use two sets of x-values–one set that approaches 1 from the left and one set that approaches 1 from the right, as shown in the table.
An Introduction to Limits
The graph of f is a parabola that has a gap at the point (1, 3), as shown in the Figure 1.5.
Although x can not equal 1, you can move arbitrarily close to 1, and as a result f(x) moves arbitrarily close to 3.
Using limit notation, you can write
An Introduction to Limits
This is read as “the limit of f(x) as x approaches 1 is 3.”
Figure 1.5
This discussion leads to an informal definition of a limit:
A limit is the value (meaning y value) a function approaches as x approaches a particular value from the left and from the right.
Properties of Limits:
Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power.
For a limit to exist, the function must approach the same value from both sides.
One-sided limits approach from either the left or right side only.
x clim f x L
x clim f x L
x clim f x L
The limit of a function refers to the value that the function approaches, not the actual value (if any).
2
limx
f x
not 1
2
y f x
1 2 3 4
1
2
At x=1: 1
lim 0x
f x
1
lim 1x
f x
1 1f
left hand limit
right hand limit
value of the function
1
limxf x
does not exist because the left and right hand limits do not match!
1limxf x
DNE
y f x
At x=2: 2
lim 1x
f x
2
lim 1x
f x
2 2f
left hand limit
right hand limit
value of the function
2
lim 1x
f x
because the left and right hand limits match.
1 2 3 4
1
2
2
lim 1xf x
y f x
At x=3: 3
lim 2x
f x
3
lim 2x
f x
3 2f
left hand limit
right hand limit
value of the function
3
lim 2xf x
because the left and right hand limits match.
1 2 3 4
1
2
3
lim 2xf x
y f x
You Try– Estimating a Limit Numerically
Use the table feature of your graphing calculator to
evaluate the function at several
points near x = 0 and use the results to estimate the limit:
Example 1 – Solution
The table lists the values of f(x) for several x-values near 0.
0
lim 2xf x
Example 1 – Solution
From the results shown in the table, you can estimate the limit to be 2.
This limit is reinforced by the graph of f (see Figure 1.6.)
cont’d
Figure 1.6
Use a graphing utility to estimate the limit:
3 2
1
1lim
1x
x x x
x
Find
2
1
2
limx
f x
1
1, 20, 2xxf x
2
limx
f x
Limits That Fail to Exist
Show that does not exist
1
1
Non-existance
x
f xx
0limx
x
x
Because the behavior differs from the right and from the left of zero, the limit DNE.
Discuss the existence of the limit:
Solution: Using a graphical representation, you can see that x does not approach any number. Therefore, the limit does not exist.
0limx
20
1limx x
0limx
0limxDNE
Fig. 1.10, p. 51
0limxDNE
Limits That Fail to Exist - 3 Reasons
Properties of Limits
Limits Basics
limx cx c
limx cb b
lim n n
x cx c
lim n n
x cx c
3lim 2x
2
5limxx
5
2
3limxx
9
3
8limxx
2
Examples
Properties of Limits
Using Properties of Limits
3
limx
f x g x
1
2
3limx
g x
3
limx
f x g x
5
84
2 3
3 3
lim 7 lim 12x x
If f x and g x
• Find the following limits:
2
4limxx
4lim5x
x
9limx
x
lim cosxx x
16
3
20
2
1lim 6x
x x
6
2
5
3 10lim
5x
x x
x
7
Properties of Limits
Compute the following limits
2
1lim
1x x
3
3lim
1x
x
x
sin 2limx
x
x
21
1lim
2x
x
x x
4
lim tanx
x
lim cosxx x
• Let’s take a look at the last one
• What happened when we plugged in 1 for x?
• When we get we have what’s called an
indeterminate form
• Let’s see how we can solve it
2
1lim
21
xx
xx
0
0
• Let’s look at the graph of
Is the function continuous at x = 1?
2
1lim
21
xx
xx
You Try:
Find the limit:
5
You Try:
3
3
27Evaluate lim
3x
x
x
27
Find the limit:
Solution:
By direct substitution, you obtain the indeterminate form 0/0.
Example – Rationalizing Technique
In this case, you can rewrite the fraction by rationalizing the numerator.
cont’dSolution
Now, using Theorem 1.7, you can evaluate the limit as shown.
cont’dSolution
A table or a graph can reinforce your conclusion that the
limit is . (See Figure 1.20.)
Figure 1.20
Solutioncont’d
Solutioncont’d
Group Work:Sketch the graph of f. Identify the values of c for which exists. lim
x cf x
lim exists for all values except where 4.x cf x c
Homework
p.54 8-25 all
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