U (The Squeeze Theorem) · The Squeeze Theorem U) ‹ (The Squeeze Theorem) Min Eun Gi :

The Squeeze Theorem

압착정리(The Squeeze Theorem)

Min Eun Gi : https://www.facebook.com/mineungimath

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



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


f (x) = L) , L− ε < f (x) < L + ε


h(x) = L) , L− ε < g(x) < L + ε

δ = min{δ0, δ1, δ2}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


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


h(x) = L) , L− ε < g(x) < L + ε

δ = min{δ0, δ1, δ2}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


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


h(x) = L) , L− ε < g(x) < L + ε

δ = min{δ0, δ1, δ2}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


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


h(x) = L) , L− ε < g(x) < L + ε

δ = min{δ0, δ1, δ2}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


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


h(x) = L) , L− ε < g(x) < L + ε

δ = min{δ0, δ1, δ2}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


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


h(x) = L) , L− ε < g(x) < L + ε

δ = min{δ0, δ1, δ2}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


f (x) = L)

, L− ε < f (x) < L + ε


h(x) = L) , L− ε < g(x) < L + ε

δ = min{δ0, δ1, δ2}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


f (x) = L) , L− ε < f (x) < L + ε


h(x) = L) , L− ε < g(x) < L + ε

δ = min{δ0, δ1, δ2}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


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}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


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}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


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}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


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}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


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}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


f (x) = L) , L− ε < f (x) < L + ε


h(x) = L)

, L− ε < g(x) < L + ε

δ = min{δ0, δ1, δ2}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


f (x) = L) , L− ε < f (x) < L + ε


h(x) = L) , L− ε < g(x) < L + ε

δ = min{δ0, δ1, δ2}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


f (x) = L) , L− ε < f (x) < L + ε


h(x) = L) , L− ε < g(x) < L + ε

δ

= min{δ0, δ1, δ2}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


f (x) = L) , L− ε < f (x) < L + ε


h(x) = L) , L− ε < g(x) < L + ε

δ = min{δ0, δ1, δ2}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


f (x) = L) , L− ε < f (x) < L + ε


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



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


f (x) = L) , L− ε < f (x) < L + ε


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



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


f (x) = L) , L− ε < f (x) < L + ε


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



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


f (x) = L) , L− ε < f (x) < L + ε


h(x) = L) , L− ε < g(x) < L + ε

δ = min{δ0, δ1, δ2}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


f (x) = L) , L− ε < f (x) < L + ε


h(x) = L) , L− ε < g(x) < L + ε

δ = min{δ0, δ1, δ2}


∴ ∀ε > 0 , ∃δ > 0

s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


f (x) = L) , L− ε < f (x) < L + ε


h(x) = L) , L− ε < g(x) < L + ε

δ = min{δ0, δ1, δ2}


∴ ∀ε > 0 , ∃δ > 0 s.t.

0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


f (x) = L) , L− ε < f (x) < L + ε


h(x) = L) , L− ε < g(x) < L + ε

δ = min{δ0, δ1, δ2}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ

⇒ |g(x)− L| < ε



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


f (x) = L) , L− ε < f (x) < L + ε


h(x) = L) , L− ε < g(x) < L + ε

δ = min{δ0, δ1, δ2}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒

|g(x)− L| < ε



The Squeeze Theorem

Start


x→af (x) = lim

x→ah(x) = L

limx→a

g(x) = L

Proof.ε > 0


f (x) = L) , L− ε < f (x) < L + ε


h(x) = L) , L− ε < g(x) < L + ε

δ = min{δ0, δ1, δ2}


∴ ∀ε > 0 , ∃δ > 0 s.t.0 < |x− a| < δ ⇒ |g(x)− L| < ε



The Squeeze Theorem

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