Limits & continuity -solutions HW %

Limits & continuity -solutions HW % - handa (nh5757)...

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Unformatted text preview: handa (nh5757) Limits & continuity bormashenko (54880) 1 This print-out should have 16 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Below is the graph of a function f . 2 4 6- 2- 4- 6 2 4 6 8- 2- 4 Use the graph to determine all the values of x on (- 6 , 6) at which f fails to be continuous. 1. x =- 3 2. no values of x 3. x = 4 4. x =- 3 , 4 correct 5. none of the other answers Explanation: Since f ( x ) is defined for all values of x on (- 6 , 6), the only values of x in (- 6 , 6) at which the function f is discontinuous are those for which lim x x f ( x ) = f ( x ) or lim x x- f ( x ) = l i m x x + f ( x ) . The only possible candidates here are x =- 3 and x = 4. But at x =- 3 f (- 3) = 8 = lim x - 3 f ( x ) = 4 , while at x = 4 lim x 4- f ( x ) = 6 = lim x 4+ f ( x ) = 4 . Consequently, on (- 6 , 6) the function f fails to be continuous only at at x =- 3 , 4 . 002 10.0 points Use continuity to evaluate lim x 3 sin ( x + 4 sin x ) . 1. limit = 2. limit = 0 correct 3. limit = 3 4. limit = 1 5. limit =- 1 Explanation: Because both x are sin x is continuous on (- , ), the sum x + 4 sin x also is contin- uous everywhere on (- , ). But then the composition f ( x ) = sin( x + 4 sin x ) too is continuous everywhere on (- , ). Now by definition, lim x c f ( x ) = f ( c ) whenever f is continuous at x = c . For the given function f , therefore, lim x 3 f ( x ) = sin(3 + 4 sin(3 )) . Consequently, limit = 0 . handa (nh5757) Limits & continuity bormashenko (54880) 2 003 10.0 points Find all the points at which the function f ( x ) = x 2- 2 x- 8 is not continuous. 1. x = 4 2. x =- 2 3. continuous everywhere correct 4. none of these 5. x = 4 ,- 2 Explanation: Since f ( x ) = x 2- 2 x- 8 is a polynomial function, it is continuous everywhere , including the points x = 4 ,- 2 at which f (4) = f (- 2) = 0 . 004 10.0 points Find all values of x at which the function f defined by f ( x ) = x- 6 x 2- x- 30 is continuous, expressing your answer in in- terval notation. 1. (- , 6) (6 , ) 2. (- ,- 5) (- 5 ,- 6) (- 6 , ) 3. (- ,- 6) (- 6 , 5) (5 , ) 4. (- ,- 5) (- 5 , ) 5. (- ,- 5) (- 5 , 6) (6 , ) correct Explanation: After factorization the denominator be- comes x 2- x- 30 = ( x- 6)( x + 5) , so f can be rewritten as f ( x ) = x...
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Limits & continuity -solutions HW % - handa (nh5757)...

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