Cypress College Math Review: Epsilon-Delta Proofs of Limits

limf ( x )  L means that for every   0 , there exists   0 such that xc xc implies that f() x L 

xc means that x is within  of c

1. Find the limit. Even if the limit is given to you, check that you wrote the problem down correctly. Most of the proofs that you will be doing will involve linear or quadratic functions. For both of these type of functions, to find the limit you simply substitute in the value of c. limf ( x )  f c xc 2. Have a copy of a proof that your instructor did as a format for your own proof. Instructors vary on how they want these proofs formatted. What is right is the way YOUR instructor wants the proof formatted NOT the way someone else tells you how to do it. You CAN learn from others the math involved, but you must use your instructor’s format for your proof.

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Find the limit L. Then use the  definition to prove that the limit is L. 4 Example) limx  1 x63

Example) lim 3 2x x4

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2 Example) limxx 3 5 x2 

Example)

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2 Example) limxx 3 x1 

Example)

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Extra Practice – Try these on your own, then check with the answers below. Find the limit L. Then use the  definition to prove that the limit is L. 2 1. limx  7 x33 1 2. lim 9  x x42 2 3. limxx 5 3 x2  2 4. limxx 5 x2 

Answers – The correct “answer” is a well written proof that matches the format laid out by your instructor. Here are simply the values of delta. 3 1.  2 2.  2  3.   min 1, 10  4.   min 1, 6

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