limits

# limits - NOTES ON LIMITS In these notes we give the...

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NOTES ON LIMITS In these notes we give the definition of the limit of a function ; the material on se- quences is in Appendix D.2 of the book. Our goal is to see how to use this definition to prove some of the types of statements which you will see in Math 104. We are not trying to fully explain what a limit actually is, and so we give no motivation nor further insight into the definition; you will see all of this in Math 104. Again, here we just want to see some basic examples of proofs involving ’s and δ ’s. Definition 1. Let f : R R be a function. We say that L R is the limit of f as x approaches a R if for every > 0, there exists δ > 0 such that | f ( x ) - L | < if 0 < | x - a | < δ . We denote this by writing lim x a f ( x ) = L . Proposition 1. Let f : R R be a constant function, f ( x ) = c . Then for any a R , lim x a f ( x ) = c Proof. Let a R . Let > 0 and let δ = 1. Then if 0 < | x - a | < δ , we have | f ( x ) - c | = | c - c | = 0 < . Hence lim x a f ( x ) = c . Proposition 2. Let f : R R be the function defined by f ( x ) = x + 3 . Then lim x 1 f ( x ) = 4 .

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