MAT 137 LEC 0601 Instructor: Alessandro Malus TA: Julia Kim - - PowerPoint PPT Presentation

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Oct 31, 2022 207 likes •299 views

MAT 137 LEC 0601 Instructor: Alessandro Malus TA: Julia Kim October 2nd, 2020 Warm-up question : Let a R , and let f and g be positive functions defined near a (except possibly at a ). Assume that f ( x ) x a f ( x ) = lim lim x

SLIDE 1

MAT 137 — LEC 0601

Instructor: Alessandro Malusà TA: Julia Kim October 2nd, 2020 Warm-up question: Let a ∈ R, and let f and g be positive functions defined near a (except possibly at a). Assume that lim

x→a f (x) = lim x→a g(x) = 0. What can we conclude about lim x→a

f (x) g(x)?

1 The limit is 1. 2 The limit is 0. 3 The limit is ∞. 4 The limit does not exist. 5 We do not have enough

information to decide.

SLIDE 2

A theorem about limits

Let f be a function with domain R such that lim

x→0 f (x) = 3

Prove that lim

x→0

5f (2x) = 15

directly from the definition of limit. Do not use any of the limit laws.

SLIDE 3

A theorem about limits

Let f be a function with domain R such that lim

x→0 f (x) = 3

Prove that lim

x→0

5f (2x) = 15

directly from the definition of limit. Do not use any of the limit laws.

1 Write down the formal definition of the statement you want to prove.

SLIDE 4

A theorem about limits

Let f be a function with domain R such that lim

x→0 f (x) = 3

Prove that lim

x→0

5f (2x) = 15

directly from the definition of limit. Do not use any of the limit laws.

1 Write down the formal definition of the statement you want to prove. 2 Write down what the structure of the formal proof should be, without

filling the details.

SLIDE 5

A theorem about limits

Let f be a function with domain R such that lim

x→0 f (x) = 3

Prove that lim

x→0

5f (2x) = 15

directly from the definition of limit. Do not use any of the limit laws.

1 Write down the formal definition of the statement you want to prove. 2 Write down what the structure of the formal proof should be, without

filling the details.

3 Rough work.

SLIDE 6

A theorem about limits

Let f be a function with domain R such that lim

x→0 f (x) = 3

Prove that lim

x→0

5f (2x) = 15

directly from the definition of limit. Do not use any of the limit laws.

1 Write down the formal definition of the statement you want to prove. 2 Write down what the structure of the formal proof should be, without

filling the details.

3 Rough work. 4 Write down a complete proof.

SLIDE 7

Proof feedback

1 Is the structure of the proof correct?

(First fix ε, then choose δ, then ...)

2 Did you say exactly what δ is? 3 Is the proof self-contained?

(I do not need to read the rough work)

4 Are all variables defined? In the right order? 5 Do all steps follow logically from what comes before?

Do you start from what you know and prove what you have to prove?

6 Are you proving your conclusion or assuming it?

SLIDE 8

Rational limits

Consider the function h(x) = (x − 1)(2 + x) x2(x − 1)(2 − x).

Find all real values a for which h(a) is undefined.
For each such value of a, compute lim

x→a+ h(x) and lim x→a− h(x).

Based on your answer, and nothing else, try to sketch the graph of h.

SLIDE 9

Before next class...

Watch videos 2.12, 2.13.
Download the next class’s slides (no need to look at them!)