146 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



Example: 



fsinh x x cosh x~\ xsmhx~\ 



n = \- 



L sin x x cos* Jz=o L 



[sinh *"[ _ |~ cosh *"[ 

 sin x J x=0 ~ cos x ] x = 



7.104 Failure of L'Hospital's Rule. In certain cases this rule fails to determine 

 the true value of an expression for the reason that all the higher derivatives 

 vanish at the limit. In such cases the true value may often be found by factoring 

 the given expression, or resolving into partial fractions ( 1.61) . 

 Example : 



7.105 In applying L'Hospital's Rule, if any of the successive quotients contains 

 a factor which can be evaluated at once its determinate value may be substituted. 



Example: 



[(i - x)e x - i "1 -xe x 



tan 2 x J X =Q \_2 tan x sec 2 X] X =Q 





|_tan 

 Hence the given function is, 



T- _i -I 



I 



2 



7.106 If the given function can be separated into factors each of which is 

 indeterminate, the factors may be evaluated separately. 

 Example : 



'(e x -i) tan 2 *", .. . =j 



z=0 



7.110 ~. If, for * = a, ^* . takes the form -^, this quotient may be 

 written: 



which takes the form - for x = a and the preceding sections will apply to it. 

 7.111 L'Hospital's Rule (7.102) may be applied directly to indeterminate forms 



CO 



, if the expansion by Taylor's Theorem is valid. 



