148 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



'ir . , 



^/ J*=0 



(x) and # approach o 

 limit, then, 



7.141 If f(x) and x approach oo together, and ^ . * approaches a definite 



J( x ) 



Limit 



vifl _ Limit f(x + i) 

 -*->co /(*) ' 



7.150 Differential Coefficients of the form -. In determining the differential 

 coefficient -j- from an equation f(x, y) = o, by means of the formula, 



/ 



^ = -- d) 



dx df (I) 



dy 

 it may happen that for a pair of values, x = a, y = b, satisfying f(x, y) = o, 



dy o 



-j- takes the form -. 

 dx o 



dy 

 Writing -g = y', and applying 7.102 to the quotient (i), a quadratic equation 



is obtained for determining y', giving, in general, two different determinate values. 

 If y' is still indeterminate, apply 7.102 again, giving a cubic equation for deter- 

 mining y'. This process may be continued until determinate values result. 



Example: 



/(*, y) = (x* + y*) 2 - c*xy = o, 



, _ 4*(* 2 + /) ~ c*y 

 * z -c 2 x' 



For # = o, y = o, y' takes the value - 

 Applying 7.102, 



-/ = 



Solving this quadratic equation in y 1 ', the two determinate values, y' = o, y f = oo ; 

 result for x = o, y = o. 



