The usual ε − δ definition of limits, Definition. limx→a f(x) = L exactly when for...

The usual ε − δ definition of limits, Definition. limx→a f(x) = L exactly when for every ε > 0 there is a δ > 0 such that for any x with |x − a| < δ we are guaranteed to have |f(x) − L| < ε as well.

1. Use the ε − δ definition of limits to verify that limx→1 (−2x + 1) = −1. [2]

2. Use the definition of limits that you didn’t use in answering question 1 to verify that limx→2 (−x + 2) does not =1. [2]

3. Use either definition of limits above to verify that limx→3 (x^2− 5)= 2. [3] Hint: The choice of δ in 3 will probably require some slightly indirect reasoning. Pick some arbitrary smallish positive number for δ as a first cut. If it doesn’t do the job, but x is at least that close, you’ll have more information to help pin down the δ you really need. Note: The problems above are probably easiest done by hand, though Maple and its competitors do have tools for solving inequalities which could be useful.

5. Compute limx→0 sin (x + π)/x by hand. [1

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Colby Messinger answered 1 year ago

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