MAT 137 LEC 0601 Instructor: Alessandro Malus TA: Muhammad Mohid - - PowerPoint PPT Presentation

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MAT 137 LEC 0601 Instructor: Alessandro Malus TA: Muhammad Mohid October 1st, 2020 Warm-up question : Suppose you need to prove that x a f ( x ) = L . lim 1 Write down the formal definition of the statement; 2 Write down what the

SLIDE 1

MAT 137 — LEC 0601

Instructor: Alessandro Malusà TA: Muhammad Mohid October 1st, 2020 Warm-up question: Suppose you need to prove that lim

x→a f (x) = L . 1 Write down the formal definition of the statement; 2 Write down what the structure of the formal proof should be.

This is for your reference: please wait before sharing your answer.

SLIDE 2

Preparation: choosing deltas

1 Find one value of δ > 0 such that

|x − 3| < δ = ⇒ |5x − 15| < 1.

2 Find all values of δ > 0 such that

|x − 3| < δ = ⇒ |5x − 15| < 1.

3 Find all values of δ > 0 such that

|x − 3| < δ = ⇒ |5x − 15| < 0.1.

4 Let us fix ε > 0. Find all values of δ > 0 such that

|x − 3| < δ = ⇒ |5x − 15| < ε.

SLIDE 3

What is wrong with this “proof"?

Prove that lim

x→3(5x + 1) = 16

“Proof:"

Let ε > 0. WTS ∀ε > 0, ∃δ > 0 s.t. 0 < |x − 3| < δ = ⇒ |(5x + 1) − (16)| < ε |(5x + 1) − (16)| < ε ⇐ ⇒ |5x + 15| < ε ⇐ ⇒ 5|x + 3| < ε = ⇒ δ = ε 3

SLIDE 4

Your first ε − δ proof

Goal

We want to prove that lim

x→3 (5x + 1) = 16

directly from the definition.

SLIDE 5

Your first ε − δ proof

Goal

We want to prove that lim

x→3 (5x + 1) = 16

directly from the definition.

1 Write down the formal definition of the statement (??).

SLIDE 6

Your first ε − δ proof

Goal

We want to prove that lim

x→3 (5x + 1) = 16

directly from the definition.

1 Write down the formal definition of the statement (??). 2 Write down what the structure of the formal proof should be, without

filling the details.

SLIDE 7

Your first ε − δ proof

Goal

We want to prove that lim

x→3 (5x + 1) = 16

directly from the definition.

1 Write down the formal definition of the statement (??). 2 Write down what the structure of the formal proof should be, without

filling the details.

3 Write down a complete formal proof.

SLIDE 8

A harder proof

Goal

We want to prove that lim

x→0

x3 + x2

= 0 directly from the definition.

SLIDE 9

A harder proof

Goal

We want to prove that lim

x→0

x3 + x2

= 0 directly from the definition.

1 Write down the formal definition of the statement (??).

SLIDE 10

A harder proof

Goal

We want to prove that lim

x→0

x3 + x2

= 0 directly from the definition.

1 Write down the formal definition of the statement (??). 2 Write down what the structure of the formal proof should be, without

filling the details.

SLIDE 11

A harder proof

Goal

We want to prove that lim

x→0

x3 + x2

= 0 directly from the definition.

1 Write down the formal definition of the statement (??). 2 Write down what the structure of the formal proof should be, without

filling the details.

3 Rough work: What is δ?

SLIDE 12

A harder proof

Goal

We want to prove that lim

x→0

x3 + x2

= 0 directly from the definition.

1 Write down the formal definition of the statement (??). 2 Write down what the structure of the formal proof should be, without

filling the details.

3 Rough work: What is δ? 4 Write down a complete formal proof.

SLIDE 13

Is this proof correct?

Claim: ∀ε > 0, ∃δ > 0 s.t. 0 < |x| < δ = ⇒ |x3 + x2| < ε.

Proof:

Let ε > 0.
Take δ =
ε

|x + 1|.

Let x ∈ R. Assume 0 < |x| < δ. Then

|x3 + x2| = x2|x + 1| < δ2|x + 1| = ε |x + 1||x + 1| = ε.

I have proven that |x3 + x2| < ε.

SLIDE 14

Choosing deltas again

Let us fix numbers A, ε > 0. Find:

1 a value of δ > 0

s.t. |x| < δ = ⇒ |Ax2| < ε

SLIDE 15

Choosing deltas again

Let us fix numbers A, ε > 0. Find:

1 a value of δ > 0

s.t. |x| < δ = ⇒ |Ax2| < ε

2 all values of δ > 0

s.t. |x| < δ = ⇒ |Ax2| < ε

SLIDE 16

Choosing deltas again

Let us fix numbers A, ε > 0. Find:

1 a value of δ > 0

s.t. |x| < δ = ⇒ |Ax2| < ε

2 all values of δ > 0

s.t. |x| < δ = ⇒ |Ax2| < ε

3 a value of δ > 0

s.t. |x| < δ = ⇒ |x + 1| < 10

SLIDE 17

Choosing deltas again

Let us fix numbers A, ε > 0. Find:

1 a value of δ > 0

s.t. |x| < δ = ⇒ |Ax2| < ε

2 all values of δ > 0

s.t. |x| < δ = ⇒ |Ax2| < ε

3 a value of δ > 0

s.t. |x| < δ = ⇒ |x + 1| < 10

4 all values of δ > 0

s.t. |x| < δ = ⇒ |x + 1| < 10

SLIDE 18

Choosing deltas again

Let us fix numbers A, ε > 0. Find:

1 a value of δ > 0

s.t. |x| < δ = ⇒ |Ax2| < ε

2 all values of δ > 0

s.t. |x| < δ = ⇒ |Ax2| < ε

3 a value of δ > 0

s.t. |x| < δ = ⇒ |x + 1| < 10

4 all values of δ > 0

s.t. |x| < δ = ⇒ |x + 1| < 10

5 a value of δ > 0

s.t. |x| < δ = ⇒

|Ax2| < ε

|x + 1| < 10

SLIDE 19

Choosing deltas again

Let us fix numbers A, ε > 0. Find:

1 a value of δ > 0

s.t. |x| < δ = ⇒ |Ax2| < ε

2 all values of δ > 0

s.t. |x| < δ = ⇒ |Ax2| < ε

3 a value of δ > 0

s.t. |x| < δ = ⇒ |x + 1| < 10

4 all values of δ > 0

s.t. |x| < δ = ⇒ |x + 1| < 10

5 a value of δ > 0

s.t. |x| < δ = ⇒

|Ax2| < ε

|x + 1| < 10

6 a value of δ > 0

s.t. |x| < δ = ⇒ |(x + 1)x2| < ε

SLIDE 20

Before next class...

Watch videos 2.5, 2.6.
Download the next class’s slides (no need to look at them!)

SLIDE 21

Before next class...

Watch videos 2.5, 2.6.
Download the next class’s slides (no need to look at them!)