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Unformatted text preview: handa (nh5757) – Limits & continuity – bormashenko – (54880) 1 This printout should have 16 questions. Multiplechoice 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 π sin ( x + 4 sin x ) . 1. limit = ∞ 2. limit = 0 correct 3. limit = 3 π 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 π f ( x ) = sin(3 π + 4 sin(3 π )) . 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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This note was uploaded on 11/21/2011 for the course M 408N taught by Professor Gualdini during the Spring '10 term at University of Texas.
 Spring '10
 Gualdini
 Continuity, Limits

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