L'Hopital's Rule and Limits

Date: 7 Aug 1995 08:59:44 -0400
From: Greg Sharpe
Subject: Limits

Find   Lim  [sin 3x  /  tan (x/3)]
       x->0

Date: 7 Aug 1995 09:54:14 -0400
From: Dr. Ken
Subject: Re: Limits

Hello!

The best tool you can use in this situation is L'Hopital's rule, which says
that as long as the numerator and denominator are both going to zero (or
both going to positive infinity, or both going to negative infinity), we can
take the derivative of the numerator and the denominator.  Doing that, we get

3 * Cos(3x)     9 * Cos(3x)
___________  =  ___________
Sec^2(x/3)/3     Sec^2(x/3)

Plugging in zero to this fraction, the Cosine and the Secant terms both go
to one, so the whole fraction goes to 9.  

-K