L`Hopital`s Rule

rational functions, l`hopital`s rule, limits, derivatives

L'Hopital's Rule is a very important rule that helps one find the limits of functions fairly easily.   While one can abusively use the rule to find limits of functions even before a section on calculus about the rule is introduced, it is appropriately used after understanding the concept of derivatives.

L'Hopital's Rule states that if the limit of a rational function give us the ratio 0/0, we can keep taking the derivative of the numberator and the denominator functions of the rational function until a limit that exists is reached.

so, if lim     f(x)           0
        x->a---------- =  ---  
                  g(x)          0

Then, L'Hopital's Rule says that we can find the limit by doing:

so, if lim     f'(x)        
        x->a----------
                  g'(x)    

and we would get the same limit.  Thus, it means that we can keep taking the derivative of the numerator and then the denominator independent of one another each time, since for f'(x), f''(x) is only its first derivative though it is the second derivative of f'(x).  In this manner, we will reach a point where the limit will be known.

For instance: If f(x)= (x^3-x^2-11x+15) and g(x)= (x-3), then, the we can have a function h(x)=f(x)/g(x).  Now, when we try to find the limit of h(x), we see that we get the following.

so, if lim     (x^3-x^2-11x+15)        
        x->3   ------------------------  =    
                            (x-3)                      

   (3^3-3^2-11(3)+15)       27-9-33+15     0
   ------------------------  =   ------------------ =---
             (3-3)                             0               0


Now, we can do what L'Hopital's Rule says and take the derivative of the numerator and the denominator to get:

lim              (3x^2-2x-11)        
        x->3   ------------------------  =    
                            1                      


             (3(3)^2-2(3)-11)        3(9)-6+-11        27-17
        ------------------------  =    ---------------- = ------------
                        1                              1                   1
=10

So, we see that the limit is 10, and we were able to do this using L'hopital's rule and this is how it works for all of our purposes of finding limits.

  

rational functions, l`hopital`s rule, limits, derivatives