January 9th, 2011, 09:37 AM  #1 
Newbie Joined: Jan 2011 Posts: 3 Thanks: 0  Limit to Infinity
Hi, Next week I have a calculus exam, so I tried some old exams. There is one question I just cant solve: Limit[(2x^26^x)/((3x+2^x)(3^xx)),x>infinity] I used mathematica to get the answer so I know it is 1 but I have no idea why... Can anyone explain it to me 
January 9th, 2011, 09:58 AM  #2 
Newbie Joined: Jan 2011 Posts: 22 Thanks: 0  Re: Limit to Infinity 
January 9th, 2011, 10:08 AM  #3  
Newbie Joined: Jan 2011 Posts: 3 Thanks: 0  Re: Limit to Infinity Quote:
 
January 9th, 2011, 10:20 AM  #4 
Newbie Joined: Jan 2011 Posts: 22 Thanks: 0  Re: Limit to Infinity
Sorry if that link wasn't very helpful, try this one http://www.calculushelp.com/howdo...aluatelimits/ 
January 9th, 2011, 10:25 AM  #5 
Math Team Joined: Apr 2010 Posts: 2,778 Thanks: 361  Re: Limit to Infinity
Hello Stef, Or: Because . Same calculations can be written included the other terms in the numerator of the limit (which is ) Got it? Hoempa 
January 9th, 2011, 11:25 AM  #6 
Senior Member Joined: Nov 2010 From: Staten Island, NY Posts: 152 Thanks: 0  Re: Limit to Infinity
Let me give you a quick shortcut: Just take the term with the fastest growing function on top and divide by the term with the fastest growing function on bottom. Then take the limit. This is Remarks: (1) We needed to multiply out the denominator in order to find the fastest growing term. (2) An exponential function with base greater than 1 grows faster than any polynomial function. 
January 9th, 2011, 02:04 PM  #7  
Newbie Joined: Jan 2011 Posts: 3 Thanks: 0  Re: Limit to Infinity Quote:
Maybe more questions will follow for this exam, is kinda sucks... Thanks *Solved*  
January 9th, 2011, 05:43 PM  #8 
Senior Member Joined: Nov 2010 From: Staten Island, NY Posts: 152 Thanks: 0  Re: Limit to Infinity
Yes. That is essentially correct as long as those are the only types of terms that appear. This will also work if there are logarithms and/or roots. Keep in mind that exponential functions grow faster than polynomials which grow faster than logarithmic functions. Also a function with a root can be treated like a polynomial by using fractional exponents. The idea is that as x gets larger the faster growing function takes over, and all the slower growing ones become negligible. So "in the limit" the slower growing ones disappear completely (this is of course a very informal argument). I would recommend that you make sure that you are allowed to use these shortcuts on your exams. I think that most teachers are ok with them (they always give the correct answer after all), but some may want you to show the details in which case you'll only get partial credit. 

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