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Unformatted text preview: Name : PID : TA : Sec. No : Sec. Time : Math20C Midterm 2. February 25 2011 Turn off and put away your cell phone. No calculators or any other electronic devices are allowed during this exam. You may use one page of notes, but no books or other assistance during the exam. Read each question carefully, and answer each question completely. Show all of your work; no credit will be given for unsupported answers. Write all your solutions clearly and legibly; no credit will be given for illegible solutions. If any question is not clear, ask for clarification. # Points Score 1 6 2 6 3 6 4 6 5 6 Σ 30 Problem 1 (6 points): Explain why the following two limits do not exist. (a) lim ( x,y ) → (0 , 0) 5 xy x 2 + y 2 . Solution: Approaching the origin along the line y = mx , the limit is lim x → 5 x ( mx ) x 2 + ( mx ) 2 = lim x → 5 mx 2 (1 + m 2 ) x 2 = 5 m (1 + m 2 ) . Since the limit depends on the direction from which we approach (0 , 0), then the limit does not exist. (b) lim ( x,y ) → (0 , 0) x 2 y x 4 + y 2 . Solution: Approaching the origin along the line y = x , the limit is lim x → x 2 ( x ) x 4 + x 2 = lim x → x x 2 + 1 = 0 . But approaching the origin along the parabola y = x 2 , the limit is lim x → x 2 ( x 2 ) x 4 + x 4 = 1 2 ....
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 Winter '08
 Helton
 Math, Calculus, lim, ... ..., Gradient

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