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September 2nd, 2012, 02:41 AM  #1 
Member Joined: Aug 2012 Posts: 88 Thanks: 0  How to find lim as x > inf. of sin(x) algebraically
Given: I understand that sin(x) oscillates between 1 and 1 as we go towards infinity in either direction, but if I didn't know what the graph looked like or the general behavior of sin(x), how would I write algebraically to show that the limit doesn't exist? The work I show is usually plugging in large numbers close to each other to show that the yvalues of sin(x) oscillate, but I don't think that is a valid way of doing this unless I use like 10 different xvalues and plug them all in. 
September 2nd, 2012, 03:27 AM  #2 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: How to find lim as x > inf. of sin(x) algebraically [color=#000000]Suppose that , then for every sequence such that , it holds that . But choosing and , so the limit does not exist. [/color] 
September 2nd, 2012, 07:45 AM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1123 Math Focus: Elementary mathematics and beyond  Re: How to find lim as x > inf. of sin(x) algebraically
Perhaps of interest, 
September 2nd, 2012, 07:51 AM  #4  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: How to find lim as x > inf. of sin(x) algebraically Quote:
 
September 2nd, 2012, 08:33 AM  #5 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1123 Math Focus: Elementary mathematics and beyond  Re: How to find lim as x > inf. of sin(x) algebraically
Claim: Let f(x) be a continuous function bounded from above and below by a and b, with both a and b finite, for all x. Then, 
September 2nd, 2012, 08:36 AM  #6  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1123 Math Focus: Elementary mathematics and beyond  Re: How to find lim as x > inf. of sin(x) algebraically Quote:
 
September 2nd, 2012, 08:38 AM  #7  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: How to find lim as x > inf. of sin(x) algebraically Quote:
By squeeze theorem, the result follows.  

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