Solving epsilon-delta problems

Solving epsilon - delta problemsMath 1A, 313,315 DISS eptember 29, 2014 There will probably be at least one epsilon - delta problem on the midterm and the kind of problems ask you to show1thatlimx af(x) =Lfor some particularfand particularL,using the actual definition of limits in terms of sand srather than the limit laws. For example, there might be a question asking you toshow thatlimx a7x+ 3 = 7a+ 3(1)orlimx 5x2 x 1 = 19,(2)using the definition of a The rules of the gameNormally, the answer to this kind of question will be of the following form:Given >0, let = [something positive, usually depending on anda]. If0<|x a|< then [some series of steps goes here], so|f(x) L|< .Some examples of this are Examples 2-4 of section Note that [some series of steps goeshere] should consist of a proof that|f(x) L|< , from the assumptions that >0 is whatever we said it was, and 0<|x a|<.

3 Strategies for nding delta One general strategy is to try solving jf(x) Lj< for x. Once you know what values of x will work, you choose so that the interval (a ;a+ ) sits inside the set of solutions. For example, suppose you’re trying to prove that lim x!8 3 p x= 2. Given >0, you need to nd >0 such that 0 <jx 8j< =)j3 p x 2j< :

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  Strategies, Delta, Epsilon, Epsilon delta

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