The usual ? ? ? definition of limits, Definition. limx?a f(x) =L exactly when for every ? > 0 there is a ? > 0 such that forany x with |x ? a| < ? we are guaranteed to have |f(x) ? L|
1. Use the ? ? ? definition of limits to verify that limx?1 (?2x+ 1) = ?1. [2]
2. Use the definition of limits that you didn’t use in answeringquestion 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 ? in 3 will probably requiresome slightly indirect reasoning. Pick some arbitrary smallishpositive number for ? as a first cut. If it doesn’t do the job, butx is at least that close, you’ll have more information to help pindown the ? you really need. Note: The problems above are probablyeasiest done by hand, though Maple and its competitors do havetools for solving inequalities which could be useful.
5. Compute limx?0 sin (x + ?)/x by hand. [1
