Unformatted text preview: h that the condition f x
V
will hold whenever x S
a , a
a . Since the interval a , a is a neighborhood of
a, condition a must hold.
7. Given that S is a set of real numbers, that f : S R, that is a real number and that a is an interior point of S,
prove that the following conditions are equivalent:
a. f x as x a. b. For every number 0 there exists a number 0 such that the inequality f x
every number x that satisfies the inequality x a  .  will hold for The only way in which condition b differs from the , form of the assertion that f x
as x a is
that it requires that f x  for all numbers within a distance of a and unequal to a. It does
not merely assert that f x  when x is a member of the set S unequal to a and within a
distance of a.
It is obvious that condition b implies condition a. To show that condition a implies condition b we
assume that f x
as x a. Suppose that 0. Choose a number 1 0 such that the
condition f x  will hold whenever x S
a 1, a 1
a . Now, using the fact that a is
an interior point of S, choose a number 2 0 such that a 2 , a 2
S. We define to be the
smaller of the two numbers 1 and 2 and we observe that the inequality f x  will hold for
every number x that satisfies the inequality x a  .
8. Suppose that S is a set of real numbers, that a is a limit point of S, that f : S R and that is a real number.
Prove that if f x
as x a then f x    as x a. Compare this exercise with an earlier exercise.
The key to this exercise is the fact that if x is any number in S then
f x    f x .
To show that f x    as x a, suppose that 0. Choose 0 such that the condition
a , a
a . Then for all such numbers x we have
f x  will hold whenever x S
f x    f x  .
9. Suppose that S is a set of real numbers, that a is a limit point of S, that f : S R and that is a real number.
Complete the following sentence: The function f fails to have a limit of at the number a when there exists a
number 0 such that for every number 0 ......
The function f fails to have a limit of at the number a when there exists a number 0 such that
for every number 0 there is at least one number x in the set S
a , a
a for which 188 f x  . 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....
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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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