L’Hospital Rule

 

               We will check one of the applications of Mean Value Theorem.

 

 

 

 

Theorem ( L’Hospital Rule :  and  )

                   Let  f(x)  and  g(x)  be continuous function and differentiable in the neighborhood of the point x=c .

(1)    If   f(c)=0  and   g(c)=0  ,  then

                                                  =

                                                                       as long as   exists .

(2)  if   f(x) =  and  g(x)=  ,   then

                                                 =    if the limit exists.

 

Proof:

                  

             (1)

                     f(c) = g(c) = 0.  Then

 

                                 =  =  =

 

                              where  a is between x and c,  b is also between x and c .

 

                     When  x  c,   we also have  a  c and b  c .    Thus,

                                   

                                       =     if the limit exists.

 

         (2)

                            f(x) =   and   g(x)=  .  Thus,

 

                            =   ,  it turns out to be the case in (1) .

                   Find the derivative of numerator and denominator:

                                   =  =

                       So,

                                  =

                           =    

                             It can be obtained   by  “cross multiplication” of the equality above ( the limit of both sides exist so that it is allowed to do that ).

 

 

 

                 Basically, the theorem just states :   when you try to find the limit of the form  or   ,  if the limit exists,  you can just find it via the limit via the quotient of the derivative functions of numerator and denominator.

 

 

 

 

 

 

Example :   Determine if     exists.

           Sol:

                            =

                          

                    So,

                                   = 0     = 0 .

 

 

 

 

 

Example:  Determine if  exists with a fixed n .

          Sol:

                           f(x)  = xⁿ

                           g(x) = e^x

 

                  We can apply  L’Hospital rule (n+1) times by finding   f⁽ⁿ⁺¹⁾(x) and g⁽ⁿ⁺¹⁾(x) :

 

                        f⁽ⁿ⁾(x) = n!  ,   f⁽ⁿ⁺¹⁾(x) = 0 ,  and g⁽ⁿ⁺¹⁾(x) = e^x

 

                 So,

                              =  = ...=  =  =0