Handout (Limits and Continuity) (Schwartz; Math 157) Work the problems on your own paper just like a normal homework assignment. 1. Sketch a possible graph of () given that lim+ () = 2, lim () = 5, and (1) = 0. 1 1 ||; 2 and use it find each of the following limits. Then ; > 2 check your answers by finding the limits numerically. (a) lim (()) 2. Sketch a graph of () = { 2 (b) lim+ (()) 2 (c) lim(()) 2 (d) lim (()) 2 3. For each of the following, find the left- and right-hand limits at the given value of x then determine whether the function is continuous at that value of x. For those that are not continuous, state what the problem is in terms of limits. +2 (a) () = |+2| at = 2. +2 (b) () = |+2| at = 2. 2; 3 at = 3 + 9; > 3 2 +5 ; 0 (d) () = { at = 0 0; = 0 (c) () = { 4. You should have found that () = { 2; 3 is not continuous at = 3. Pick a value + 9; > 3 2; 3 for C so that () = { is continuous at = 3. + ; > 3 ; 0 2 5. You should have found that () = { +5 is not continuous at = 0. However, you 0; = 0 could make it continuous if you define (0) to be something other than zero. What would (0) have to be in order to make () continuous? 2 ; 3 6. Find values for m and n in the function () = { so that lim () = 5. Then + ; > 3 3 sketch the graph of (). Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner