October 6th, 2012, 11:10 AM  #1 
Senior Member Joined: Jun 2011 Posts: 154 Thanks: 0  2 questions (limits/derivatives)
Hello everyone. I have a few problems I am stumped on. Studying for a test on Monday and need some guidance. Here are the problems: 1) Limit as x approaches 0 of (5x) / tan(7x). Can't plug in 0, as it will result in a 0 on the denominator. So I multiply by the reciprocal, which will be (5x) * [cos(x)/sin(x)]. And can not think of what to do next. Is it ok to plug the 0 into the denominator of the recriprocal, which would result in a 1 in the denominator, and leave us with 5x * cos(7x)? Or is that not the correct way to go about it. I don't know why but I always get stuck on these trig limit problems. 2) Given that f(x) = 3xh(x), h(2) = 1, and hprime(2) = 1 Find fprime(2) I know how to find derivatives I just don't know exactly how to go about this problem. Should I plug 1 in for x first? In fact while I'm at it I might as well ask for suggestions on the best way to go about solving this: find dy/dx of [(6x^2  1/x)]*(3x+1) I am familiar with the product rule, quotient rule, and with how to find derivatives. I just know that there are sometimes easier ways than others to find them. For the problem above, I end up with a mass of jumbled numbers, letters, and exponents. I am hoping someone can tell me how the simplest way to do it. 
October 6th, 2012, 11:21 AM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: 2 questions (limits/derivatives)
1.) Let 2.) 
October 6th, 2012, 11:47 AM  #3  
Senior Member Joined: Jun 2011 Posts: 154 Thanks: 0  Re: 2 questions (limits/derivatives) Quote:
Quote:
[/quote] Not sure if you noticed but I did edit my post so if anyone can help me there is a 3rd problem in my original post I am also stuck on.  
October 6th, 2012, 12:12 PM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: 2 questions (limits/derivatives)
If you know L'Hôpital's rule (most times it seems students haven't learned this yet, that's why I didn't use it), since you mention differentiation, then you could simply write: For your third problem, I would write: Now, using the product rule, we find: 
October 6th, 2012, 12:30 PM  #5  
Senior Member Joined: Jun 2011 Posts: 154 Thanks: 0  Re: 2 questions (limits/derivatives) Quote:
Quote:
 
October 6th, 2012, 12:36 PM  #6 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: 2 questions (limits/derivatives)
Yes, the coefficient of 7 comes from the application of the chain rule as .

October 6th, 2012, 12:40 PM  #7  
Senior Member Joined: Jun 2011 Posts: 154 Thanks: 0  Re: 2 questions (limits/derivatives) Quote:
Oh and I still am not quite sure about problem 2. Not sure if you noticed I typed my message into the quote below: Quote:
[/quote] What process/rule was done to solve this problem? And also, as I noted above, in the 2nd part What happened to the x in the 2nd term? Why is the x gone but not the h?  
October 6th, 2012, 12:49 PM  #8 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: 2 questions (limits/derivatives)
I used the product rule: 
October 6th, 2012, 12:56 PM  #9  
Senior Member Joined: Jun 2011 Posts: 154 Thanks: 0  Re: 2 questions (limits/derivatives) Quote:
 
October 6th, 2012, 03:21 PM  #10 
Senior Member Joined: Jun 2011 Posts: 154 Thanks: 0  Re: 2 questions (limits/derivatives)
............. EDIT 

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