Calculus: LimitsRigorouslyDefined by: javier AWS
Calculus: Limits Rigorously Defined
by: javier
The Limit Definition understanding the definition
The Limit Definition: understanding the definition ( ) 3 Consider lim 1 − x→∞ x
deterime the best predition for the y value. The expected limit value might be 1, but □ ... does the function ever get within .5 of 1?
1
0.5
0
−0.5
∞ 0
10
20
30
∞
The Limit Definition: understanding the definition ( ) 3 Consider lim 1 − x→∞ x
deterime the best predition for the y value. The expected limit value might be 1, but □ ... does the function ever get within .5 of 1?
1
if so, when? prove it! 0.5
0
−0.5
∞ 0
10
20
30
∞
The Limit Definition: understanding the definition ( ) 3 Consider lim 1 − x→∞ x
deterime the best predition for the y value. The expected limit value might be 1, but 1
0.5
□ ... does the function ever get within .25 of 1?
if so, when? prove it! 0
−0.5
∞ 0
10
20
30
∞
The Limit Definition: understanding the definition ( ) 3 Consider lim 1 − x→∞ x
deterime the best predition for the y value. The expected limit value might be 1, but 1
0.5
□ ... does the function ever get within .25 of 1?
if so, when? prove it! 0
−0.5
∞ 0
10
20
30
∞
The Limit Definition: understanding the definition ( ) 3 Consider lim 1 − x→∞ x
deterime the best predition for the y value. The expected limit value might be 1, but 1
0.5
0
□ ... does the function ever get within .0005 of 1?
if so, when? prove it! −0.5
∞ 0
10
20
30
∞
The Limit Definition: understanding the definition ( ) 3 Consider lim 1 − x→∞ x
deterime the best predition for the y value. The expected limit value might be 1, but 1
0.5
0
□ ... does the function ever get within .0005 of 1?
if so, when? prove it! −0.5
∞ 0
10
20
30
∞
The Limit Definition: understanding the definition ( ) 3 Consider lim 1 − x→∞ x
deterime the best predition for the y value. The expected limit value might be 1, but 1
0.5
0
−0.5
∞ 0
10
20
30
∞
□ ... does the function ever get within ϵ of 1?
if so, when? prove it!
The Limit Definition: understanding the definition ( ) 3 Consider lim 1 − x→∞ x
deterime the best predition for the y value. The expected limit value might be 1, but 1
0.5
0
−0.5
∞ 0
10
20
30
∞
□ ... does the function ever get within ϵ of 1?
if so, when? prove it!
The Limit Definition lim f(x) = L (the limit definition at ∞)
x→∞
The Limit Definition lim f(x) = L (the limit definition at ∞)
x→∞
lim f(x) = L
x→∞
if and only if ∀ϵ > 0 ∃ M such that x > M =⇒ |f(x) − L| < ϵ
The Limit Definition: lim f(x) = L (the limit definition at c) x→c
Consider lim 2x + 1 x→5
deterime the best predition for the y value. The expected limit value might be 11, but □ ... does the function ever get within 2 of 11? 10
5
0
0
1
2
3
4
5
6
The Limit Definition: lim f(x) = L (the limit definition at c) x→c
Consider lim 2x + 1 x→5
deterime the best predition for the y value. The expected limit value might be 11, but □ ... does the function ever get within 2 of 11?
if so, when? prove it! 10
5
0
0
1
2
3
4
5
6
The Limit Definition: lim f(x) = L (the limit definition at c) x→c
Consider lim 2x + 1 x→5
deterime the best predition for the y value. The expected limit value might be 11, but
10
□ ... does the function ever get within 1 of 11?
if so, when? prove it! 5
0
0
1
2
3
4
5
6
The Limit Definition: lim f(x) = L (the limit definition at c) x→c
Consider lim 2x + 1 x→5
deterime the best predition for the y value. The expected limit value might be 11, but
10
□ ... does the function ever get within 1 of 11?
if so, when? prove it! 5
0
0
1
2
3
4
5
6
The Limit Definition: lim f(x) = L (the limit definition at c) x→c
Consider lim 2x + 1 x→5
deterime the best predition for the y value. The expected limit value might be 11, but
10
□ ... does the function ever get within .05 of 11?
5
if so, when? prove it! 0
0
1
2
3
4
5
6
The Limit Definition: lim f(x) = L (the limit definition at c) x→c
Consider lim 2x + 1 x→5
deterime the best predition for the y value. The expected limit value might be 11, but
10
□ ... does the function ever get within .05 of 11?
5
if so, when? prove it! 0
0
1
2
3
4
5
6
The Limit Definition: lim f(x) = L (the limit definition at c) x→c
Consider lim 2x + 1 x→5
deterime the best predition for the y value. The expected limit value might be 11, but
10
5
0
0
1
2
3
4
5
□ ... does the function ever get within ϵ of 11? 6
if so, when? prove it!
The Limit Definition: lim f(x) = L (the limit definition at c) x→c
Consider lim 2x + 1 x→5
deterime the best predition for the y value. The expected limit value might be 11, but
10
5
0
0
1
2
3
4
5
□ ... does the function ever get within ϵ of 11? 6
if so, when? prove it!
The Limit Definition lim f(x) = L
x→c
The Limit Definition lim f(x) = L
x→c
lim f(x) = L
x→c−
if and only if ∀ϵ > 0 ∃ δ > 0 such that 0 < |x − c| < δ =⇒ |f(x) − L| < ϵ
The Limit Definition Famous Limit Properties
The Limit Definition: Famous Limit Properties
□ Limit of a constant [LK]: lim K = K
x→c
The Limit Definition: Famous Limit Properties
□ Limit of a constant [LK]: lim K = K
x→c
□ Limit of x [LX]: lim x = c
x→c
The Limit Definition: Famous Limit Properties
□ Limit of a constant [LK]: lim K = K
x→c
□ Limit of x [LX]: lim x = c
x→c
□ Limit of the sum [LS=SL]*: lim (f + g) = lim (f) + lim (g)
x→c
x→c
x→c
The Limit Definition: Famous Limit Properties
□ Limit of a constant [LK]:
□ Limit of the product [LP=SP]*:
lim K = K
lim (f · g) = lim (f) · lim (g)
x→c
x→c
□ Limit of x [LX]: lim x = c
x→c
□ Limit of the sum [LS=SL]*: lim (f + g) = lim (f) + lim (g)
x→c
x→c
x→c
x→c
x→c
The Limit Definition: Famous Limit Properties
□ Limit of a constant [LK]:
□ Limit of the product [LP=SP]*:
lim K = K
lim (f · g) = lim (f) · lim (g)
x→c
x→c
□ Limit of x [LX]:
x→c
□ Limit of the product [LQ=QL]*:
lim x = c
x→c
lim
□ Limit of the sum [LS=SL]*:
x→c
lim (f + g) = lim (f) + lim (g)
x→c
x→c
x→c
x→c
* each must exists and be finite, no zeros in denominator
f limx→c f = g limx→c g
The Limit Definition How to PROVE the famous Limit Properties
The Limit Definition: How to PROVE the famous Limit Properties
□ Limit of the sum [LS=SL]*: lim (f + g) = lim (f) + lim (g)
x→c
x→c
x→c
The Limit Definition: How to PROVE the famous Limit Properties
□ Limit of the sum [LS=SL]*:
limx→c (g) = M then prove that
lim (f + g) = lim (f) + lim (g)
x→c
x→c
x→c
in other words, if limx→c (f) = L and
lim (f + g) = L + M
x→c
The Limit Definition How to USE the famous Limit Properties
The Limit Definition: How to USE the famous Limit Properties
□ User Limit properties to PROVE That lim (3x2 + 1) = 76
x→5
by: javier
The Limit Definition understanding the definition
The Limit Definition: understanding the definition ( ) 3 Consider lim 1 − x→∞ x
deterime the best predition for the y value. The expected limit value might be 1, but □ ... does the function ever get within .5 of 1?
1
0.5
0
−0.5
∞ 0
10
20
30
∞
The Limit Definition: understanding the definition ( ) 3 Consider lim 1 − x→∞ x
deterime the best predition for the y value. The expected limit value might be 1, but □ ... does the function ever get within .5 of 1?
1
if so, when? prove it! 0.5
0
−0.5
∞ 0
10
20
30
∞
The Limit Definition: understanding the definition ( ) 3 Consider lim 1 − x→∞ x
deterime the best predition for the y value. The expected limit value might be 1, but 1
0.5
□ ... does the function ever get within .25 of 1?
if so, when? prove it! 0
−0.5
∞ 0
10
20
30
∞
The Limit Definition: understanding the definition ( ) 3 Consider lim 1 − x→∞ x
deterime the best predition for the y value. The expected limit value might be 1, but 1
0.5
□ ... does the function ever get within .25 of 1?
if so, when? prove it! 0
−0.5
∞ 0
10
20
30
∞
The Limit Definition: understanding the definition ( ) 3 Consider lim 1 − x→∞ x
deterime the best predition for the y value. The expected limit value might be 1, but 1
0.5
0
□ ... does the function ever get within .0005 of 1?
if so, when? prove it! −0.5
∞ 0
10
20
30
∞
The Limit Definition: understanding the definition ( ) 3 Consider lim 1 − x→∞ x
deterime the best predition for the y value. The expected limit value might be 1, but 1
0.5
0
□ ... does the function ever get within .0005 of 1?
if so, when? prove it! −0.5
∞ 0
10
20
30
∞
The Limit Definition: understanding the definition ( ) 3 Consider lim 1 − x→∞ x
deterime the best predition for the y value. The expected limit value might be 1, but 1
0.5
0
−0.5
∞ 0
10
20
30
∞
□ ... does the function ever get within ϵ of 1?
if so, when? prove it!
The Limit Definition: understanding the definition ( ) 3 Consider lim 1 − x→∞ x
deterime the best predition for the y value. The expected limit value might be 1, but 1
0.5
0
−0.5
∞ 0
10
20
30
∞
□ ... does the function ever get within ϵ of 1?
if so, when? prove it!
The Limit Definition lim f(x) = L (the limit definition at ∞)
x→∞
The Limit Definition lim f(x) = L (the limit definition at ∞)
x→∞
lim f(x) = L
x→∞
if and only if ∀ϵ > 0 ∃ M such that x > M =⇒ |f(x) − L| < ϵ
The Limit Definition: lim f(x) = L (the limit definition at c) x→c
Consider lim 2x + 1 x→5
deterime the best predition for the y value. The expected limit value might be 11, but □ ... does the function ever get within 2 of 11? 10
5
0
0
1
2
3
4
5
6
The Limit Definition: lim f(x) = L (the limit definition at c) x→c
Consider lim 2x + 1 x→5
deterime the best predition for the y value. The expected limit value might be 11, but □ ... does the function ever get within 2 of 11?
if so, when? prove it! 10
5
0
0
1
2
3
4
5
6
The Limit Definition: lim f(x) = L (the limit definition at c) x→c
Consider lim 2x + 1 x→5
deterime the best predition for the y value. The expected limit value might be 11, but
10
□ ... does the function ever get within 1 of 11?
if so, when? prove it! 5
0
0
1
2
3
4
5
6
The Limit Definition: lim f(x) = L (the limit definition at c) x→c
Consider lim 2x + 1 x→5
deterime the best predition for the y value. The expected limit value might be 11, but
10
□ ... does the function ever get within 1 of 11?
if so, when? prove it! 5
0
0
1
2
3
4
5
6
The Limit Definition: lim f(x) = L (the limit definition at c) x→c
Consider lim 2x + 1 x→5
deterime the best predition for the y value. The expected limit value might be 11, but
10
□ ... does the function ever get within .05 of 11?
5
if so, when? prove it! 0
0
1
2
3
4
5
6
The Limit Definition: lim f(x) = L (the limit definition at c) x→c
Consider lim 2x + 1 x→5
deterime the best predition for the y value. The expected limit value might be 11, but
10
□ ... does the function ever get within .05 of 11?
5
if so, when? prove it! 0
0
1
2
3
4
5
6
The Limit Definition: lim f(x) = L (the limit definition at c) x→c
Consider lim 2x + 1 x→5
deterime the best predition for the y value. The expected limit value might be 11, but
10
5
0
0
1
2
3
4
5
□ ... does the function ever get within ϵ of 11? 6
if so, when? prove it!
The Limit Definition: lim f(x) = L (the limit definition at c) x→c
Consider lim 2x + 1 x→5
deterime the best predition for the y value. The expected limit value might be 11, but
10
5
0
0
1
2
3
4
5
□ ... does the function ever get within ϵ of 11? 6
if so, when? prove it!
The Limit Definition lim f(x) = L
x→c
The Limit Definition lim f(x) = L
x→c
lim f(x) = L
x→c−
if and only if ∀ϵ > 0 ∃ δ > 0 such that 0 < |x − c| < δ =⇒ |f(x) − L| < ϵ
The Limit Definition Famous Limit Properties
The Limit Definition: Famous Limit Properties
□ Limit of a constant [LK]: lim K = K
x→c
The Limit Definition: Famous Limit Properties
□ Limit of a constant [LK]: lim K = K
x→c
□ Limit of x [LX]: lim x = c
x→c
The Limit Definition: Famous Limit Properties
□ Limit of a constant [LK]: lim K = K
x→c
□ Limit of x [LX]: lim x = c
x→c
□ Limit of the sum [LS=SL]*: lim (f + g) = lim (f) + lim (g)
x→c
x→c
x→c
The Limit Definition: Famous Limit Properties
□ Limit of a constant [LK]:
□ Limit of the product [LP=SP]*:
lim K = K
lim (f · g) = lim (f) · lim (g)
x→c
x→c
□ Limit of x [LX]: lim x = c
x→c
□ Limit of the sum [LS=SL]*: lim (f + g) = lim (f) + lim (g)
x→c
x→c
x→c
x→c
x→c
The Limit Definition: Famous Limit Properties
□ Limit of a constant [LK]:
□ Limit of the product [LP=SP]*:
lim K = K
lim (f · g) = lim (f) · lim (g)
x→c
x→c
□ Limit of x [LX]:
x→c
□ Limit of the product [LQ=QL]*:
lim x = c
x→c
lim
□ Limit of the sum [LS=SL]*:
x→c
lim (f + g) = lim (f) + lim (g)
x→c
x→c
x→c
x→c
* each must exists and be finite, no zeros in denominator
f limx→c f = g limx→c g
The Limit Definition How to PROVE the famous Limit Properties
The Limit Definition: How to PROVE the famous Limit Properties
□ Limit of the sum [LS=SL]*: lim (f + g) = lim (f) + lim (g)
x→c
x→c
x→c
The Limit Definition: How to PROVE the famous Limit Properties
□ Limit of the sum [LS=SL]*:
limx→c (g) = M then prove that
lim (f + g) = lim (f) + lim (g)
x→c
x→c
x→c
in other words, if limx→c (f) = L and
lim (f + g) = L + M
x→c
The Limit Definition How to USE the famous Limit Properties
The Limit Definition: How to USE the famous Limit Properties
□ User Limit properties to PROVE That lim (3x2 + 1) = 76
x→5
