Tutorial02-solutions(1)

Find 0 0 and 0 0 s 0 3 0 we have 0 0 and

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Unformatted text preview: (cos3 θ + sin3 θ ) → 0+ , we get ) = lim (cos3 θ + sin3 θ ) →0 4 )= 2+ 2 , we Also, since −1 ≤ cos3 θ ≤ 1 we have −1 ≤ sin3 θ ≤ 1 >0 −2 ≤ (cos3 θ + sin3 θ ) ≤ 2 and by the Squeeze Theorem, lim (cos3 θ + sin3 θ ) = 0 →0 E Let ( | )= |. Find (0 0) and (0 0). S (0 3 0)] | We have, ( 0) = 0 and hence (0 0) = [ ( ) = 0 and hence (0 0) = [ (0 )] | =0 = 0 | =0 = 0. = 0| = 0. Similarly, :) Questions on logical thinking =0 =0 2 E Let ( )= 2+ 4 . (a) Prove that the limit of ( (b) Prove that lim( S = )→(0 0) ) as ( ( ) approaches (0 0) along any straight line is 0, ) is, however, not deﬁned. (a) An arbitrary line passing through (0 0) can be represented either as or as = 0. We have, 2 lim ( →0 for = ) = lim →0 3 2 4 + . In the same manner, for lim (0 →0 (b) Consider the parabola = 2 4 2 = lim →0 1+ 4 2 =0 = 0 we get ) = lim 0 = 0 →0 . The limit along this parabola is 4 lim ( →0 2 ) = lim →0 4 + 4 = 1 2 On the other hand, if the limit of ( ) was equal to...
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This document was uploaded on 02/10/2014.

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