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Sharon
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 Posted: Sun Jul 13, 2008 3:10 am Post subject: Limit - is this true? |
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I'm not sure if this claim is true:
Let f and g be functions from [0, oo) to R such that
a) lim x --> oo g(x) = 0
b) for every x in [0, oo), f(x) + g(x) is in [a, b], a and b in R.
Then, that there exists M such that f(x) is in [a, b] for every x > M.
Does anything change if we add the assumptions that (1) f and g are continuous and (2) are differentiable?
Thank you.
Sharon |
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Arturo Magidin
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| Posted: Sun Jul 13, 2008 5:03 am Post subject: Re: Limit - is this true? |
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In article <7894057.1215918661730.JavaMail.jakarta@nitrogen.mathforum.org>,
Sharon <anamerryl4@yahoo.com> wrote:
| Quote: |
I'm not sure if this claim is true:
Let f and g be functions from [0, oo) to R such that
a) lim x --> oo g(x) = 0
b) for every x in [0, oo), f(x) + g(x) is in [a, b], a and b in R.
Then, that there exists M such that f(x) is in [a, b] for every x > M.
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It's not true: pick your favorite g which approaches 0 from above
(that is, g(x)>0 for every sufficiently large x), and pick your
favorite a and b. Then set f(x) = b+g(x). Thus, f(x)>b for every
sufficiently large x.
However, it ->is<- true that, under the assumptions (a) and (b), for
every e>0 there exists M such that f(x) is in (a-e,b+e)]. This because
for every e>0 there exists M>0 such that g(x) is in (-e,e). Therefore,
since a <= f(x)+g(x) < f(x) + e, we have f(x) > a-e; and likewise
from f(x)-e < f(x)+g(x) <= b we get f(x)<b+e. Thus, a-e < f(x) < b+e.
| Quote: |
Does anything change if we add the assumptions that (1) f and g are
continuous and (2) are differentiable?
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No; as you see with the example above, f can be chosen to be at least
as nice as g is chosen to be.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin
magidin-at-member-ams-org |
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The World Wide Wade
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| Posted: Sun Jul 13, 2008 8:40 am Post subject: Re: Limit - is this true? |
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In article
<7894057.1215918661730.JavaMail.jakarta@nitrogen.mathforum.org>,
Sharon <anamerryl4@yahoo.com> wrote:
| Quote: |
I'm not sure if this claim is true:
Let f and g be functions from [0, oo) to R such that
a) lim x --> oo g(x) = 0
b) for every x in [0, oo), f(x) + g(x) is in [a, b], a and b in R.
Then, that there exists M such that f(x) is in [a, b] for every x > M.
Does anything change if we add the assumptions that (1) f and g are
continuous and (2) are differentiable?
Thank you.
Sharon
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What if f(x) = -g(x)? Then f(x) + g(x) is in [0, 0] for all x, but ... |
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