Limit ProofDate: 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/ |
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