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Limit Proof


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 
contradiction.

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 
contradiction or if you have more in-depth questions about how the 
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 
check your answer or need more clues, please do write back!  

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

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