What is wrong with this method to find limit?
<br /="/"/><div><br /="/"/>What is (2^0.5)^(2^0.5)^(2^0.5)...?<br /="/"/><br /="/"/>Here is a solution.<span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/>Let it be x, then we get x=(2^0.5)^x. It seems it will lead x=2 or 4.<br /="/"/><br /="/"/>The limit should be unique. What is wrong with this method?<br /="/"/><br /="/"/>Someone said that if x^x^x... (x>0) has a finite limit, x should <br /="/"/>be <=e^(1/e). If he was right, then the maximum limit of x^x^x... <br /="/"/>is e. So 4 cannot be the answer and 2 is. Is there a way to get <br /="/"/>the answer directly (which is 2) and not use the maximum limit e to <br /="/"/>reject 4? <br /="/"/></div><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div>回复：What is wrong with this method to find limit?
<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>I think you have to make sure that the limit is approaching 2 from below so that you can reject the answer of 4. Here is the proof for </div><div>(2^0.5) (2^0.5)^(2^0.5)... < 2 for finite terms.</div><div> <span style="display:none;">www.ddhw.com</span></div><div>1. since (2^0.5) < 2</div><div>2. then (2^0.5)^(2^0.5) < (2^0.5)^2=2</div><div>3. and (2^0.5)^(2^0.5)^(2^0.5) < (2^0.5)^2 = 2</div><div>4. further more</div>(2^0.5)^(2^0.5)^(2^0.5) ... < 2 for <div> </div><div>So answer 4 should be rejected by using your method, which is mathematically sound.</div><div> </div><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>thanks a lot. What I do not understand is how
<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>it got an extra solution 4. If you have some idea, please let me know. I posted this one in wxc and Dr. math. I have not got good answers.</div><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>回复：What is wrong with this method to find limit?
<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>The limit is infinite.</div><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>回复：回复：What is wrong with this method to find limit
<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>the 2nd post says it is <=2.</div><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>回复：回复：回复：What is wrong with this method to find li
<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>That proof does not seems to be right. 1,2,3 are OK, but how 4 is achieved? </div><br /="/"/><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/><table width="100%"><tr><td colspan="10"></td><td width="100%"><i>原贴：</i><hr /="/"/><span>文章来源: yma16<span>®</span> 于 2009-12-26 11:10:30 (北京时间: 2009-12-27 0:10:30)<br /="/"/>标题：<b>回复：回复：What is wrong with this method to find limit</b><br /="/"/><br /="/"/><table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>the 2nd post says it is <=2.</div><br /="/"/><br /="/"/> </td></tr></table><hr /="/"/></span></td></tr></table><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>回复：回复：回复：回复：What is wrong with this method to find
<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>Can you think of 1,2,3 as steps in a "mathematical induction" procedure? And 4 is the conclusion. Good luck.</div><div> </div><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>It is an "irrational" number!
<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>Let me try to answer it. </div><div>I guess we can blame the "irrationality" of the number for the strange solution of 4 you got. </div><div>(2^0.5) is an irrational number. You are dealing with the irrational power of an irrational number. What is the meaning of an irrational power of an irrational number ? Can it be rational or irrational?<span style="display:none;">www.ddhw.com</span></div><div>I guess the answer is both. It is very strange. Very often, when we turn an infinite series into a finite algebraic equation to solve, we inevitably "added" something to the original problem. You have to rely on somthing else, being physical meaning or math reasoning, to get the right answer. </div><div>I hope this is the case. Let us know you have something else to explain for the "4". <span style="display:none;">www.ddhw.com</span></div><div><span style="LINE-HEIGHT: 115%; FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 11pt; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: SimSun; mso-fareast-theme-font: minor-fareast; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA"> <span style="LINE-HEIGHT: 115%; FONT-FAMILY: 'Calibri','sans-serif'; FONT-SIZE: 11pt; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: SimSun; mso-fareast-theme-font: minor-fareast; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA"> </span></span></div><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>回复：回复：回复：回复：回复：What is wrong with this method to f
<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>You are right. I think I misunderstood the order of x^x^x^x.... -- I thought I had fell in such a 'pit' once before...</div><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>回复：It is an "irrational" number!
<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>2^0.5 is the only case to lead two solutions. If you try 3^(1/3) or 5^0.2, it will not introduce an extra solution. </div><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>回复：回复：It is an "irrational" number!
<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>Maybe it has somthing to do with the fact that 2 is the only even prime number!<img /="/" src="http://www.ddhw.com/img/20.gif"></img>Xiangdang's Conjecture!</div><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>回复：What is wrong with this method to find limit?
<table cellpadding="8" height="100%" width="100%"><tr><td valign="top"><br /="/"/><div>I think the method does not work. However, I do not know why. The <br /="/"/>reason the method does not work is the following,<br /="/"/><br /="/"/>Consider x^x^x... (x>0). Replace x with 10^(1/10), using this <br /="/"/>method, we will get the result 10. But 10^(1/10) is smaller than 2^<br /="/"/>(1/2). So the result should be smaller than 2, which was the <br /="/"/>solution of (2^0.5)^(2^0.5)^(2^0.5)..., by using this method.<br /="/"/></div><span style="display:none;">www.ddhw.com</span><br /="/"/><br /="/"/> <div style="MARGIN-TOP:20px;MARGIN-LEFT:0;MARGIN-BOTTOM:0;float:left"></div></td></tr></table>
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