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Midterm1 Solutions

# Midterm1 Solutions - Midterm Examination Math 1A Professor...

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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 defined 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 k π / 2 , for k = 1 , 2 , 3 , ... . It is continuous whenever it is defined since sin and ln are continuous. That is, f is continuous except at x = (3 π / 2) + (2 π k ) for k = 1 , 2 , 3 , ... . (Note. You could have written much less than this, as long as you got the final answer and justified 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 →∞ 3 + 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

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