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Calculus Lecture 3 Notes

by: nichl

Calculus Lecture 3 Notes 01:640:135

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About this Document

Calculus I
Dr. Kontorovich
Class Notes
Calculus, Math, Limits
25 ?




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This 2 page Class Notes was uploaded by nichl on Wednesday September 14, 2016. The Class Notes belongs to 01:640:135 at Rutgers University taught by Dr. Kontorovich in Fall 2016. Since its upload, it has received 3 views. For similar materials see Calculus I in Mathematics at Rutgers University.


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Date Created: 09/14/16
Calculus 1 Lecture 3 9/13/16 Professor’s Comments  Free tutoring available at Rutgers Learning Centers  Khan Academy has videos available on most topics  Study groups are highly encouraged Limit  Loose definition: The value a function approaches but does not reach  Calculus was done for 250 years without any formal definition of a limit  Limit has nothing to do with the value of the function at a specific point, it only tells us what value the function is approaching Examples 1. ???? ???? = {4,???? > 2 function 8,???? < 2 ????→3 ????(????) = 4 Limit of function as x approaches 3 lim ????(????) = DNE Limit at 2 does not exist (see graph) ????→2 ????→2m+????(????) = 4 Exists when approached from + lim ????(????) = 8 Exists when approached from - ????→2 − The limit exists at 4 because the values on either side (i.e. 3.99 and 4.01) are both approaching the same value As x approaches 2, the function is not approaching the same value. From the negative side, the function is approaching 8. From the positive side, it is approaching 4. Therefore, the limit does not exist. If we only look at one side, however, the limit does exist. Limits only exist when you approach a real number 2. 1 ???? ???? = sin function ???? 1 ????→0 sin????= DNE Limit of function as x approaches 0 Function oscillates wildly, therefore there is no single value It approaches and therefore no limit Calculus 1 Lecture 3 9/13/16 0is the indeterminate form. This means a limit does not exist at this point, and must be 0 rewritten to determine what value the function is approaching. Note that there will be a removable discontinuity at this point 3. 1 ???? ???? = ????2 function lim 1 = ∞ (DNE) Function approaches infinity ????→0 ????2 Infinity is not a real number, so the limit technically does not exist, although we denote that it is approaching +∞ 4. √????−2 lim function ????→4 ????−4 lim √????−2 √ ????+2) Multiply by the conjugate ????→4 ????−4 (√????+2) lim ????−4 = 1 Domain: [0, 4) U (4, ∞) ????→4 (????−4)√ ????+2) 4 1 lim = DNE ????→0 √???? + 2 Rules  If lim????(????) exists and is equal to L, then lim???? = L2 ????→???? ????→????  Double f(x), double L, etc.  I????→????im????(????) = L exists an????→????im????(????) = M, then????→????m????(????) ∗ ???? ???? = L ∗ M o This is why we do not want infinity to be an actual limit, it will render the composition of limits infinity, which is not a real number and therefore not a solution ???? ????) ???? ????)  If ???? ???? = and ????(????) ≠ 0, then lim???? ???? = ???? ????) ????→???? ???? ????) Squeeze Theorem  If ℎ ???? ≤ ???? ???? ≤ ????(????) near c and lim????(????) = ???? = limℎ(????), then lim????(????) exists and is equal to ????→???? ????→???? ????→???? the value h(x) and g(x) are both approaching regardless of what f(x) does


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