MATH 1920 Lecture Notes - Lecture 10: Quotient Rule, Product Rule
15 Sep 2017
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(, ) = if for every > 0, there is a. > 0 such that if 0 < || - op || < then |f(x, y) - l| < y x. D*(p, ) = set of all points (x, y) p such that distance to p is < (shape is a punctured disc) Does the limit of f as (x, y) approaches (0,0) exist? y k = 2 k = 1 k = 1/2 x. All level curves pass through d*((0, 0), ) So, the limit does not exist for all > 0. Along y = mx, lim(x -> 0) mx/x : approaching (0, 0) along different lines gives a different limit. This implies that the limit does not exist. Key fact: in general, if the limit of a function exists, then for any two curves intersecting p, the limit to p along these curves is the same.
