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calc notes lecture 4

# calc notes lecture 4 - (f 1 1 lim x x x x f-5 Techniques...

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Lecture Sections Objective Assignment 4-5 1.5 Limits of functions 1.5: 3, 5, 13, 15, 17, 19, 23-57 eoo. Understanding Goals: 1. To understand what limit is. 2. To be able to find limits of functions graphically and numerically. 3. To use the properties of limits to evaluate limits of functions. 4. To use different analytic techniques to evaluate limits of functions. 5. To understand one-sided limits to be able to evaluate it. 6. To be able to recognized unbounded behavior of functions. Example 1 : Find the limits of the following functions both numerically and graphically as 1 x - . a) 3 1 1 x x - - Question: We know that 3 2 1 ( 1)( 1) 1 1 x x x x x x - - + + = - - , is it true that 3 2 1 1 1 x x x x - = + + - ? What is the behavior of the graph of this function near 1? x = x 0.9 0.99 0.999 0.9999 1 1.0001 1.001 1.01 1.1 f ( x ) 2.71 2.970 1 2.997 2.997001 ? 3.0003 3.003 3.0301 3.31 1

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b) | 1| ( ) 1 x f x x - = - x 0.9 0.99 0.999 0.9999 1 1.0001 1.001 1.01 1.1 f ( x ) 2
c) 2 1 ( ) 1 x f x x - = - x 0.9 0.99 0.999 0.9999 1 1.0001 1.001 1.01 1.1 f ( x ) 3

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Example 2 : Find the following limits: (a) 2 2 lim( 3 11) x x x f - + (b) 2 1 1 lim 1 x x x f - - + (c) 0 1 1 lim x x x f + - (d) ( 29 2 0 2 4 lim h h h f + - (e) 2 2 4 5 4 lim

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Unformatted text preview: + +-(f) 1 1 lim x x x x f--5 Techniques for evaluating limits: 1. Direct substitution 2. Cancellation 3. Rationalization One-sided limits: lim ( ) x c f x L + = Limit from the right lim ( ) x c f x L-= Limit from the left Example 3 : Find the limit as x f from the left and the limit as x f from the right for the function ( ) x f x x = . 6 Example 4 : Find the limit of ( ) f x as 1 x f . 2 4 , 1 ( ) 4 , 1 x x f x x x x-< =-Example5 : Find the limit (if possible) 1 1 lim 1 x x f-. 7 Example 6 : Find the limit (if possible) 2 5 lim 2 x x f-+ . Example7 : Given the following graph: Compute each of the following, a). ( 29 4 f-b). ( 29 4 lim x f x--c) ( 29 4 lim x f x +-d). ( 29 4 lim x f x f-e) ( 29 1 f f). ( 29 1 lim x f x-g) ( 29 1 lim x f x + h). ( 29 1 lim x f x f i) ( 29 6 f j). ( 29 6 lim x f x-k). ( 29 6 lim x f x + l) ( 29 6 lim x f x f 8...
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