Some Points to Note in L’hospital Rule

 

Summary of L’hospital Rule

 

Indeterminate Quotient

        0              +¥          +¥          -¥           -¥

        0              +¥          -¥           +¥          -¥

Indeterminate Product

        0×(+¥)            0×(-¥)

Indeterminate Difference

        (+¥) – (+¥)

Indeterminate Power

        00            ¥0           1¥

 

 

               Let f and g be differentiable on an interval I containing the point a (except may be point a itself) and let g’(x) ¹ 0 on  I \{a}.  If   is an indeterminate quotient as defined above, then

                            (*)

provided the right hand side limit exists (or equal to +¥, -¥).

 

Note that the rule can also be applied to one-sided limit (x®a+, x®a-) as well as infinity

(x®+¥, x®¥)

 

 

Nonexistence of RHS of (*) does not prove the nonexistence of LHS of (*)

 

Consider:

              

It is of  ¥ ¤ ¥ form, if you apply L’hospital rule (LHR), you get:

              

Since cos x does not have a limit as x®¥, the limit on RHS does not exist.

In fact, the limit L really exists, since  sin x  is bounded, we have

              

 

 

Algebra

 

       Some algebra is needed to change other indeterminate forms to the basic quotient form. In the followings the ‘0’ and ‘¥’ are “functions”.

 

Product form:      

                              

 

Difference form:

                              

Power form:

                               Take logarithm, then reduce to Product Form and then Quotient Form.

 

 

Example

 

      

(1) 

      

               which is more complicate than the original problem.

(2)

              

               \    L = e0 = 1

 

 

Interesting cases

 

(1)  Applying the L’hospital rule (LHR) below:

 

      

 

      The result is cyclic!

 

       In fact, we may evaluate as follows:

                      

(2)

      

       which cannot yield any result.

      

       But by taking logarithms, we get: