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ws05 - WORKSHEET 5 Fall 1995 1 Use a calculator to ll in...

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WORKSHEET 5 - Fall 1995 1. Use a calculator to fill in the table given and then make a guess at the given limit. a) lim x 3 x 3 x 3 = x 2.99 2 2.5 2.75 2.9 limit x - 3 x - 3 b) lim x 2 3 x 3 x + 2 = 3x + 2 x limit 1 1.5 1.75 1.9 1.99 3x c) lim x 0 1 x 2 = x 1 0.5 0.25 0.1 0.001 limit x 1 2 d) lim x 0 (1 + x ) 1 /x = x 0.0001 0.001 0.01 0.1 1 limit (1+x) 1/x e) lim x 0 sin 6 x sin 2 x = sin 6x sin 2x x 1 limit 0.001 0.0001 0.25 0.1 Describe how this technique for guessing limits could fail. 2. Evaluate the following limits algebraically (you may need to be clever for some of them.) a) lim x 3 x 3 x 3 b) lim x →− 2 x 2 + 5 x + 6 x 2 + x 2 c) lim x 4 + h 2 h d) lim x 8 3 x 2 x 8
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3. Below are some properties of limits which seem quite reasonable, yet they are not always true. For each one, state what hypotheses are necessary for the property to hold. (Note for example that if f ( x ) = 1 /x , g ( x ) = x , and b = 0 then Property 3 certainly fails.) A. Constant Multiplier Property: lim x b kf ( x ) = k lim x b f ( x ) B. Sum Property: lim x b [ f ( x ) + g ( x )] = lim x b f ( x ) + lim x b g ( x ) . C. Product Property: lim x b f ( x ) · g ( x ) = lim x b f ( x )
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