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Unformatted text preview: it follows from the deﬁnition that lim
x→a x→a (x) − l = 0 means that for any given > 0, there exists a δ > 0 such that for any 0 < x − a < δ , we
have  (x) − l − 0 < . Note that we have (x) − l ≥ 0 so that  (x) − l − 0 = (x) − l .
Therefore, we see that for any > 0, there exists a δ > 0 such that for any 0 < x − a < δ ,
we have (x) − l < , which implies that lim (x) = l by deﬁnition.
:)
x→a 2 Questions on calculation
22 E Prove that lim
( )→(0 0) 22 + ( − )2 is not deﬁned (by ﬁnding limits along dif ferent straight lines).
S Let’s ﬁnd the limits along the lines = 0 and = . For = 0, we have 22 lim (
→0 lim ( For 0) = lim ) = lim →0 2 02 0
=0
+ ( − 0)2 = , we have
2
→0 →0 2 2 2 + ( − )2 =1 Since limits along diﬀerent lines are diﬀerent, the limit of the whole function is not
deﬁned.
:)
E Find the limit if it exists, or show that the limit does not exist.
2 (a)
lim )→(2 1) ( 4−
2+3 2 (b)
4 lim )→(0 0) ( 4 +3 2 4 + (c)
lim
( )→(0 0) +2 (d)
2 +
2+ )→(0 0 0) ( +
2+ 3 lim 2 3 4 (e)
lim )→(0 0) ( S +
2+ 2 (a) (
) = 4− 2 is a rational function and as such it is continuous every2 +3
where where the denominator is nonzero. In particular, this function is continuous
at (2 1), so by continuity
lim
( ( )→(2 1) ) = (2 1) = 2
4−2·1...
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 Spring '13

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