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**Unformatted text preview: **MATH 8, SECTION 1, WEEK 4 - RECITATION NOTES TA: PADRAIC BARTLETT Abstract. These are the notes from Friday, Oct. 22nd’s lecture. In this talk, we wrap up a number of loose ends relating to continuity and limits, discussing one-sided limits, limits at infinity, the intermediate value theorem, and the concepts of open, closed, and bounded sets. 1. Random Question Question 1.1. Can you find a function f : [0 , 1] → [0 , 1] such that • f is continuous, • f (0) = 0 ,f (1) = 1 , and • f takes on every value in the interval (0 , 1) exactly once? Twice? Three times? n times? Infinitely many times? Today’s lecture is kind of a grab-bag of topics; where Monday and Wednesday’s lectures were devoted to exploring a pair of complicated topics slowly and carefully, most of the ideas in today’s lecture are relatively short and sweet. Consequently, we’ll move at a faster pace; there are about four concepts that we should cover today, each of which is hopefully a little related to the others and should be useful in your study of limits and continuity. 2. One-Sided Limits Let’s start with something fairly elementary: the concept of a one-sided limit : Definition 2.1. For a function f : X → Y , we say that lim x → a + f ( x ) = L if and only if (1) (vague:) as x goes to a from the right-hand-side, f ( x ) goes to L . (2) (concrete, symbols:) ∀ > , ∃ δ > 0 s.t. ∀ x ∈ X, ( | x- a | < δ and x > a ) ⇒ ( | f ( x )- L | < ) . Similarly, we say that lim x → a- f ( x ) = L if and only if (1) (vague:) as x goes to a from the left-hand-side, f ( x ) goes to L . (2) (concrete, symbols:) ∀ > , ∃ δ > 0 s.t. ∀ x ∈ X, ( | x- a | < δ and x < a ) ⇒ ( | f ( x )- L | < ) . 1 2 TA: PADRAIC BARTLETT Basically, this is just our original definition of a limit except we’re only looking at x-values on one side of the limit point a : hence the name “one-sided limit.” Thus, our methods for calculating these limits are pretty much identical to the methods we introduced on Monday: we work one example below, just to reinforce what we’re doing here....

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