Math 1a: Limit Laws (Solutions)
September 16, 2009
1.
Let
f
be a function such that
f
(2
.
01) = 0,
f
(2
.
001) = 0,
f
(2
.
0001) = 0, and so on. Can we conclude that
lim
x
→
2
f
(
x
) = 0?
Solution.
No, we cannot. To conclude that lim
x
→
2
f
(
x
) = 0, we must know that for
any
x
sufficiently close
to 2,
f
(
x
) is close to 0 and not just for
x
of the form 2
.
01, 2
.
001, 2
.
0001 etc.
2.
Suppose we know that lim
x
→
5
f
(
x
) = 3. Which of the following must be true?
1.
f
(5) = 3;
2.
f
is defined at 5;
3. if
f
is defined at 5 then
f
(5) = 3;
4. there is a number
x
such that
f
(
x
)
<
3
.
05; (hint: look for
x
near 5)
5. there is a number
x
such that
f
(
x
)
>
2
.
95;
Solution.
(a), (b), (c) are false. In general, in dealing with lim
x
→
a
f
(
x
), the behavior of
f
at
a
is irrelevant.
(d) and (e) are true. Since lim
x
→
5
f
(
x
) = 3, if we take any
x
sufficiently close to 5, then
f
(
x
) will be within
distance 0
.
05 of 3. That is, for an
x
sufficiently close to 5 (but not equal to 5), we have 2
.
95
< f
(
x
)
<
3
.
05.
Slightly more formally, if we take a ‘red interval’ around 3 of radius 0
.
05, then there is a ‘green interval’
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 Fall '09
 BenedictGross
 Math, Derivative, Limit, lim, Limit of a function

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