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Limit Proof

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Date: 7/13/96 at 17:34:34
From: Anonymous
Subject: Limit Proof

If lim(An/N) = L and L>0, how do you show that
lim(An) = + infinity?

An  is a function a sub n.

Barry Moore, Mathematics Teacher
Newark High School
```

```
Date: 7/15/96 at 20:16:24
From: Doctor Sydney
Subject: Re: Limit Proof

Dear Barry,

Hello!  Great question.  I'm assuming that the capital N is the same
as the lower case n, and so you know the limit as n goes to infinity
of An/n  = L>0, and you want to show that the lim as n goes to
infinity of An is positive infinity.

Well, one way you could approach the problem is to do a proof by

Suppose the limit as n goes to infinity of An is NOT positive
infinity. Then, what could it be? Well, it could be finite, or it
could be negative infinity, right? Suppose it is finite. Then, we
would have that the limit as n goes to infinity of An/n is 0, right?
This is becuase as n gets big, the numerator approaches some number,
m, but the denominator gets HUGE! But this contradicts our assumption
that the limit was L>0, so the limit as n goes to infinity of An
couldn't have been finite.

So, we consider the other case. Suppose the limit as n goes to
infinity of An is negative infinity. What does this imply about the
limit as n goes to infinity of An/n? Find a contradiction, and you
will be done! You will have then shown that the limit can't be
negative infinty or finite. Hence, the only possibility left is that
it is positive infinity, and your proof is complete.

I hope that helps. If you have any questions about that last
proof might go, please feel free to write us back. We'd also love to
hear from your students if they have any burning questions.

--Doctor Sydney,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 7/18/96 at 23:33:37
From: Anonymous
Subject: Limit Proof

I can solve the following problem:

Let f,g:R to R  be continuous at a point c, and let
h(x)m = sup{f(x),g(x)}
for all x in R. I can show that h(x) = 1/2(f(x)+g(x)) + 1/2|f(x)-g(x)|
for all x in R. I can also show that h is continuous at c.

My Question is:

What happens if I change h(x) to equal the infimum instead of the
supremum?

That is, if h(x) = inf{f(x),g(x)}, then how does this change the
problem?

Barry Moore, Mathematics Teacher
Newark High School
```

```
Date: 9/19/96 at 19:30:13
From: Doctor Sydney
Subject: Re: Limit Proof

Dear Barry:

I have a suggestion. Why don't you try to come up with an equation
for h(x) = inf{f,g} the way you did for sup. With a litte work, I bet
it can be done. Model it after the equation you got for h sup above.
From there, you'll be set for showing continuity. If you want to

-Doctor Sydney,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus

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