L'Hospital's Rule

L'Hospital's Rule is a method for finding the value of a function using derivatives. This method is use when, the value at a point don't exist.

Suppose there are continuous function f(x) and g(x) that are both zero at x=a. Then, the limit   cannot be found by substituting x=a since it give 0/0 which cannot be evaluated we use 0/0 as a notation for an expression known as an indeterminate form may be evaluated by cancellation or rearrangement of terms. However, this does not always work.
      For example, how would you evaluate   ? Obviously inserting x=0 will give an indeterminate form of 0/0 and, in this case you can neither use algebraic manipulation nor rearrangement of terms to reduce this expression into a form that yield a valid limit.

L'Hospital's Rule

Statement: If f and g are differentiable function such that

1.   or

     ;

2. f'(x), g'(x) both exist and
g'(x)≠0 ∀ x∈(a-δ,a+δ), δ>0 except possibly at x=a;

3.  exist
Then

Examples

Q1. Evaluate the limit : .
Ans.
       Directly applying x=0 leads the limit to an indeterminate form. Since both the term in the numerator and the term in the denominator are zero at x=0 and since sin x and x are both differentiable at x=0, we can use L'Hospital's Rule.

          .

Q2. Evaluate 
Ans.
                              

Directly applying x=0 leads the limit to an indeterminate form. Since both the term in the numerator and the term in the denominator are zero at x=0 and since both are differentiable at x=0, we can use L'Hospital's Rule.

    =  

    =  

                          
    = 1.

Q3. Evaluate  .
Ans.
                                

Directly applying x=0 leads the limit to an indeterminate form. Since both the term in the numerator and the term in the denominator are zero at x=0 and since both are differentiable at x=0, we can use L'Hospital's Rule.
       =                      
     
Since after applying L'Hospital's Rule we are getting 0/0 form. So we apply the rule again till we didn't get 0/0 form.

       =                       

       =                       

       =                      

       =    
       =  120×0 
       =     0