Math 106 Notes

Math 106 Notes - Calculus with Applications Dr. Cong-Cong...

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Unformatted text preview: Calculus with Applications Dr. Cong-Cong Xing Dept of Math and Computer Science l Please go to www.coursecompass.com to register if you have not done so. Homework will start soon. l Course ID: see syllabus Ch11 Differential Calculus Section11.1 Limits DEF of Limits (Definition) Idea of Limits x y L a L x f a x = ) ( lim l Ex1: Given f(x) = x. Find lim f(x). (That is, what is the limit of f(x) as x approaches 2?) 2 x l Solution: (Method 1): Draw the graph of the function (if possible) and examine the behavior of the function near x=2. x y 1 2 3 1 2 3 L a l As we can see from the graph, f(x) approaches 2 as x approaches 2 (from both sides). l what are f(x), a, and L (in the DEF of limits) in this case? Answer: f(x) =x, a =2, L =2. l Hence, the limit of f(x) as x approaches 2 is 2. In symbol, lim f(x) = 2. 2 x Method 2: Make a table showing the values of f(x) near x=2. (This method is particularly useful when graphing method does not work.) The table shows (again) that f(x) is very close to 2 when x approaches 2 from both sides. Therefore, lim f(x) = 2. x 1.97 1.98 1.99 2 2.01 2.02 2.03 f(x) 1.97 1.98 1.99 2.01 2.02 2.03 2 x l Ex2: Given Find 1 ) ( 2 + + = x x x f ) ( lim x f 3 x l Solution: Build a table showing the values of f(x) near x=3. l The table suggests that lim f(x) = 13 3 x X 2.9 2.99 2.999 3 3.001 3.01 3.1 f(x) 12.31 12.93 Fill in Fill in 13.07 13.71 l Ex3: Given Find (Note: f(x) is NOT defined at x=2, is this a problem?) 2 2 3 2 ) ( 2--- = x x x x f ) ( lim x f 2 x l Solution: Build a table showing values of f(x) near x=2. l The table suggests that lim f(x) = 5 2 x X 1.99 1.999 2 2.0001 2.001 f(x) 4.98 4.998 undef Fill in 5.002 l Ex4: Given f(x) = Find lim f(x) 1 X is an integer otherwise 4 x l Solution: We can see from the graph that when x approaches 4 from both sides, f(x) approaches 1(actually remains 1). So lim f(x) = 1. y x 1 2 3 4-4 -3 -2 -1 4 x 1 Summary (Ex 3) (ex 3) (ex 1) (ex 4) Non-Existence of Limits 2 4 ) ( 2- + = x x x g 2 x Ex4.1: Given Find Solution: Make a table showing values of g(x) near x=2 and examine them. ) ( lim x g The table shows that g(x) does NOT approach any real number when x approaches 2 from both sides. Hence, by the def of limits, lim g(x) does not exist. 2 x Ex4.2: Let Find lim f(x) Solution: The graph of f is as follows x x x f = ) ( x We can see from the graph, as x approaches 0 from the right, f(x) approaches (actually remains) 1; as x approaches 0 from the left, f(x) approaches (actually remains) -1. Since 1 -1, f(x) does not approach a single real number as x approaches 0 from both sides. Therefore, by the def of limits, lim f(x) does not exist ....
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This note was uploaded on 10/17/2011 for the course MATH 106 taught by Professor Dr.ma during the Spring '11 term at Nicholls State.

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Math 106 Notes - Calculus with Applications Dr. Cong-Cong...

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