Midterm1 Solutions - Midterm Examination Math 1A Professor J Harrison Fall 2009 Solutions(1(2 pts Where is the function f(x = ln(1 sin(x continuous

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Midterm Examination Math 1A Professor J. Harrison Fall 2009 Solutions (1) (2 pts) Where is the function f ( x ) = ln(1 + sin( x )) continuous? Answer: The function is de±ned whenever 1 + sin( x ) > 0. That is, we need sin( x ) > - 1. Since - 1 sin( x ) 1, we know sin( x ) > 1 every- where except where sin ( x )= - 1. This only occurs when x =3 kπ/ 2 , for k =1 , 2 , 3 , ... . It is continuous whenever it is de±ned since sin and ln are continuous. That is, f is continuous except at x = (3 π/ 2) + (2 πk ) for k , 2 , 3 , ... . (Note. You could have written much less than this, as long as you got the ±nal answer and justi±ed it somehow.) (2) (2 pts) Evaluate lim x →∞ 3+ cos 2 x x 2 . Answer: lim x →∞ 3= 3. Since 0 cos 2 x x 2 1 /x 2 and lim x →∞ 1 /x 2 = 0, we may apply the squeeze theorem to conclude cos 2 x x 2 = 0. By the summation limit law, we know lim x →∞ cos 2 x x 2 = 3. (3) (2 pts) Evaluate lim x →∞ 9 x 5 - 2 x 2 +1 x 4 - 3 x 3 +2 x Answer: lim x →∞ 9 x 5 - 2 x 2 +1 x 4 - 3 x 3 +2 x = lim x →∞ 9 x 5 x 4 - 2 x 2 x 4 + 1 x 4 x 4 x 4 - 3 x 3 x 4 + 2 x x 4 = lim x
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This note was uploaded on 06/26/2010 for the course MATH 1A taught by Professor Wilkening during the Fall '08 term at University of California, Berkeley.

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Midterm1 Solutions - Midterm Examination Math 1A Professor J Harrison Fall 2009 Solutions(1(2 pts Where is the function f(x = ln(1 sin(x continuous

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