Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L...
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Transcript of Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L...
![Page 1: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/1.jpg)
Chapter 2: Limits
2.2The Limit of a Function
![Page 2: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/2.jpg)
Limits
• “the limit of f(x), as x approaches a, equals L”• If we can make the values of f(x) arbitrarily
close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a) but not equal to a
Lxfax
)(lim
![Page 3: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/3.jpg)
Helpful notes…
• In limits, x ≠ a• This means we never consider that x = a
• The only thing that matters is how f(x) behaves near a
![Page 4: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/4.jpg)
Example 1
• Guess the value of 11lim 21
xx
x
![Page 5: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/5.jpg)
Example 2
• Estimate the value of 2
2
0
39limt
tt
![Page 6: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/6.jpg)
Example 3
• Guess the value of xx
x
sinlim0
![Page 7: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/7.jpg)
Example 4
• Investigate xx
sinlim0
![Page 8: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/8.jpg)
Example 5
• Find
000,10
5coslim 3
0
xxx
![Page 9: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/9.jpg)
Example 6
• The Heaviside function H is defined by
• What is the limit?
0,10,0
)(tt
tH
![Page 10: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/10.jpg)
One sided limits
• Left-hand limit of f(x) as x approaches a
• Approaches from the negative side
Lxfax
)(lim
![Page 11: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/11.jpg)
One sided limits
• Right-hand limit of f(x) as x approaches a
• Approaches from the positive side
Lxfax
)(lim
![Page 12: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/12.jpg)
Therefore…
Lxfax
)(lim
If and only if…
andLxfax
)(lim Lxfax
)(lim
![Page 13: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/13.jpg)
Example 7• The graph of a function g is shown in Figure 10
on page 71. Use it to state the values (if they exist) of the following:
)(lim
)(lim
)(lim
2
2
2
xg
xg
xg
x
x
x
![Page 14: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/14.jpg)
Example 7• The graph of a function g is shown in Figure 10
on page 71. Use it to state the values (if they exist) of the following:
)(lim
)(lim
)(lim
5
5
5
xg
xg
xg
x
x
x
![Page 15: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/15.jpg)
Example 8
• Find if it exists20
1limxx
![Page 16: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/16.jpg)
Definition
• Let f be a function defined on both sides of a, except possibly at a itself
• Then
• Means that the values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a, but not equal to a
• Happens in cases of functions with asymptotes!
)(lim xfax
![Page 17: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/17.jpg)
Definition
• Let f be a function defined on both sides of a, except possibly at a itself
• Then
• Means that the values of f(x) can be made arbitrarily large negative by taking x sufficiently close to a, but not equal to a
• Happens in cases of functions with asymptotes!
)(lim xfax
![Page 18: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/18.jpg)
Vertical Asymptotes
• The line x = a is called a vertical asymptote of the curve y = f(x) if at least one of the following statements is true:
)(lim
)(lim
)(lim
xf
xf
xf
ax
ax
ax
![Page 19: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/19.jpg)
Example 9a
• Find 3
2lim3 x
xx
![Page 20: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/20.jpg)
Example 9b
• Find 3
2lim3 x
xx
![Page 21: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/21.jpg)
Example 10• Find the vertical asymptotes of f(x) = tan x
![Page 22: Chapter 2: Limits 2.2 The Limit of a Function. Limits the limit of f(x), as x approaches a, equals L If we can make the values of f(x) arbitrarily close.](https://reader035.fdocuments.us/reader035/viewer/2022082208/5a4d1b707f8b9ab0599b4fcf/html5/thumbnails/22.jpg)
Homework
• P.74
• 4 – 9, 21, 25, 29