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2
2
2
x2 y2 2
x2 y2
x2 y2 and, by considering the cases x 
and we conclude that f x, y y  and y  x , we see that
f x, y  x  y 
0 as x, y
0, 0 . 11. Suppose that S is a set of real numbers, that a is a limit point of S, that f : S R k and that
that if f x
as x a then f x
as x a. Compare this exercise with
the corresponding exercise for sequences.
The key to this exercise is the fact that, whenever x is in the domain of f we have

fx
 fx
and the proof runs along the same lines as the one used for sequences.
12. Suppose that X is a metric space, that S is a subset X and that both of the sets S and X
Suppose that c X and that
x if x S c if x fx X R k . Prove S are dense in X. . S Prove that f has a limit at c but does not have a limit at any other point of the space X.
To show that f x
c as x c, suppose that 0. We define and observe that whenever
x c and d c, x we have
d c, f x d c, x
when x S and we have
d c, f x d c, c 0
when x X S.
Now suppose that x X
c . We want to show that f has no limit at x. To obtain a contradiction,
suppose that f has a limit at x. We define d c, x and note that 0. Choose 0 such that
/2, and such that for every point t X B c,
c , we have d f t , /4. Choose points t 1
and t 2 in the set X B x,
x such that t 1 S and t 2 X S. We see that
d x, c
d x, t 1 d t 1 , c
d t1, c
2
from which we conclude that d t 1 , c /2. Therefore
d f t1 , f t2 d t1, c
On the other hand, 215 2 . d f t1 , f t2 d f t 1 , d , f t 2 4 4 2 . This is the desired contradiction. Some Further Exercises on Limits
1. Given that
fx 1 if x 2
0 if x 2 prove that f has a limit from the left at 2 and also has a limit from the right at 2 but does not have a limit at 2.
The fact that f does not have a limit at 2 will be clear when we have seen that f x
1 as x 2
and f x
0 as x 2 . Suppose that 0. We define 3 (or just take to be any positive
number you like). Whenever x 2 and x 2  we have
f x 1  1 1  0
and whenever x 2 and x 2  we have
f x 0  0 0  0 .
2. Given that
1
x 3 
for all numbers x 3, explain why f has a limit (an infinite limit) at 3.
We need to show that f x
Ý as x 3. Suppose that w is a real number. Given x 3, the
inequality f x w says that
1
w.
x 3 
We can’t simply turn these expressions over because w need not be positive but we can make the
observation that the inequality
1
w
x 3 
will certainly hold when
1
w  1
x 3 
which says that
1.
x 3 
w  1
We therefore define w11 and observe that the condition f x w will hold whenever x 3 and

x 3  .
fx 3. Given that
1
x3
for all numbers x 3, explain why f has an infinite limit from the left at 3 and also has an infinite limit from
the right at 3 but does not have a limit at 3.
The reason f has no twosided limit at 3 is that the limits of f at 3 from the left and from the right are
not equal to each other. In fact,...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.
 Fall '08
 STAFF
 Math, Calculus

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