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. CCMR Epsilon-Delta Proofs of Limits Page 1 of 5 Find the limit L. Then use the definition to prove that the limit is L. 4 Example) limx 1 x63 Example) lim 3 2x x4 CCMR Epsilon-Delta Proofs of Limits Page 2 of 5 2 Example) limxx 3 5 x2 Example) CCMR Epsilon-Delta Proofs of Limits Page 3 of 5 2 Example) limxx 3 x1 Example) CCMR Epsilon-Delta Proofs of Limits Page 4 of 5 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. limx 7 x33 1 2. lim 9 x x42 2 3. limxx 5 3 x2 2 4. limxx 5 x2 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 CCMR Epsilon-Delta Proofs of Limits Page 5 of 5 .