The brilliant L'Hopital.

When evaluating a limit, you may find using L'Hopital's Rule helpful!  The general idea is to find the derivative of the numerator and the derivative of the denominator.  Do not use the quotient rule! Treat the numeriator as "f(x)" and the denominator as "g(x)".  then find f'(x) and g'(x).  Continue finding the derivative until it is possible to plug in the a term give in the limit.  After a is plugged in, then you will calculate the number, and that is the limit. (Lifshitz 263)

NOTE: L'Hopital's rule can only be used when the function approaches 0/0 or infinity/infinity.  

  This equation says: as the formula approaches zero, what is the limit?  We use L'Hopital's Rule to discover that it is 8.  After we found the derivatives twice, the denominator became a constant.  At this point, L'Hopital's Rule cannot be used again, instead it is time to plug in the x term.

http://www.calculusapplets.com/lhopitals.html  This applet will allow you to plug in an equation given to you and visually see.  L'hopital's rule works because it compares the ratio of the derivatives of the numerator and the denominator.  This applet will allow you visually see the ratio.

a)10

b)one-fourth

c)0

d)infinity

(purple)

(red)

  (blue)

   (green)

a)1/37

b)infinity

c)0

d)14

(orange)

joyce: i know you want to use vectors, please resist on this problem.

a) 0

b)1/4

c) 1

d) 3/5

(yellow)

1/12

infinity (if you look at the equation at first it looks like it will go to zero, this is why we need to use l'hoptial's rule)

d

0

infinity

c

I would like to thank the following people who have helped me:

my imagination for the wonderful formulas

Lifshitz's book that has wonderful calculus and definition of L'Hopital's rule

and finally I would like to thank the CalculusApplet, without you, I would be nothing.