Limits and Continuity 14 2 Math 241 Spring 2014 Jayadev S Athreya Spring 2014 Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits Limits The most important concept of calculus is the limit Informally we say lim f x L x a if no matter how close you want f x to get to L you can find a small neighborhood of size so that x being within of a guarantees that f x is within of L All this relies on is a notion of distance We have to be careful in higher dimensions because you can approach a point in many different ways Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits One variable limits The sentence limx a f x L means the following For every 0 acceptable error There is 0 how close you need to be Such that 0 x a implies f x L Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits One variable limits reprise Given h R h 6 0 let E h f a h L The sentence limx a f x L means the following For every 0 acceptable error There is 0 how close you need to be Such that 0 h implies E h Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits One sided limits In one variable a limit exists if and only if the one sided limits exist and are equal Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits One sided limits In one variable a limit exists if and only if the one sided limits exist and are equal Limit from the left limx a f x L if For every 0 acceptable error There is 0 how close you need to be Such that 0 a x implies f x L Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits One sided limits In one variable a limit exists if and only if the one sided limits exist and are equal Limit from the left limx a f x L if For every 0 acceptable error There is 0 how close you need to be Such that 0 a x implies f x L Limit from the right limx a f x L if For every 0 acceptable error There is 0 how close you need to be Such that 0 x a implies f x L Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits One sided limits reprise In one variable a limit exists if and only if the one sided limits exist and are equal Given h R h 6 0 let E h f a h L Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits One sided limits reprise In one variable a limit exists if and only if the one sided limits exist and are equal Given h R h 6 0 let E h f a h L Limit from the left limx a f x L if For every 0 acceptable error There is 0 how close you need to be Such that 0 h implies E a h Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits One sided limits reprise In one variable a limit exists if and only if the one sided limits exist and are equal Given h R h 6 0 let E h f a h L Limit from the left limx a f x L if For every 0 acceptable error There is 0 how close you need to be Such that 0 h implies E a h Limit from the right limx a f x L if For every 0 acceptable error There is 0 how close you need to be Such that 0 h implies E a h Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits Examples linear function Let s try and see that limx 3 4x 1 Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits Examples linear function Let s try and see that limx 3 4x 1 13 Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits Examples linear function Let s try and see that limx 3 4x 1 13 Given h 6 0 What is E h 4 3 h 1 13 Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits Examples linear function Let s try and see that limx 3 4x 1 13 Given h 6 0 What is E h 4 3 h 1 13 E h 4 h Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits Examples linear function Let s try and see that limx 3 4x 1 13 Given h 6 0 What is E h 4 3 h 1 13 E h 4 h Making E h small So to make E h what do we need to make h smaller than Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits Examples linear function Let s try and see that limx 3 4x 1 13 Given h 6 0 What is E h 4 3 h 1 13 E h 4 h Making E h small So to make E h what do we need to make h smaller than We get 4 Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits Examples quadratic function Let s try and see that limx 2 5x 2 1 Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits Examples quadratic function Let s try and see that limx 2 5x 2 1 21 Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits Examples quadratic function Let s try and see that limx 2 5x 2 1 21 Given h 6 0 What is E h 5 2 h 2 1 21 Jayadev S Athreya Math 241 Spring 2014 Limits and Continuity 14 2 One variable limits Higher dimensional limits Directional Limits Examples quadratic function Let s try and see that limx 2 5x 2 1 21 Given h 6 0 What is E h 5 2 h 2 1 21 E h 5h2 20h …
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