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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.
 Spring '13

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