The Squeeze Theorem U)‹ (The Squeeze Theorem) Min Eun Gi : https://www.facebook.com/mineungimath
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The Squeeze Theorem
압착정리(The Squeeze Theorem)
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theorem
f (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , limx→a
f (x) = limx→a
h(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0
s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t.
0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1
⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒
|f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε
(∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L)
, L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0
s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t.
0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2
⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒
|h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε
(∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L)
, L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ
= min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x)
, L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε
, L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε
, |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0
s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.
0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ
⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒
|g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
Start
Theoremf (x) ≤ g(x) ≤ h(x)(0 < |x− a| < δ0) , lim
x→af (x) = lim
x→ah(x) = L
limx→a
g(x) = L
Proof.ε > 0
∃δ1 > 0 s.t. 0 < |x− a| < δ1 ⇒ |f (x)− L| < ε (∵ limx→a
f (x) = L) , L− ε < f (x) < L + ε
∃δ2 > 0 s.t. 0 < |x− a| < δ2 ⇒ |h(x)− L| < ε (∵ limx→a
h(x) = L) , L− ε < g(x) < L + ε
δ = min{δ0, δ1, δ2}
f (x) ≤ g(x) ≤ h(x) , L− ε < f (x) ≤ g(x) ≤ h(x) < L + ε , L− ε < g(x) < L + ε , |g(x)− L| < ε
∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε
Min Eun Gi : https://www.facebook.com/mineungimath
The Squeeze Theorem
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Min Eun Gi : https://www.facebook.com/mineungimath
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