SPECIAL APPLICATIONS OF ANALYSIS 147 



Example : 



7.112 If /(*) and * approach oo together, and if f(x + i) - /(*) approaches a 

 definite limit, then, 



Limit r/Wl Limit 



x oo 



7.120 o X oo . . If, for x = a, f(x) x F(x) takes the form o x oo , this product 

 may be written, 



F(x) 

 which takes the form - (7.101). 



710A Tr Limit ., . . Limit _, N 



7.130 oo-oo. If f(x) =00 and F(x) = oo 



' x >a J - 



K "TFT ^ s ^ nCerent ^ rom um "ty the true value of /(#) - F(x) for a; = a is oo . 



If ./ ^ = + i, the expression has the indeterminate form oo x o which 



X-+co f( X ) 



may be treated by 7.120. 



7.140 i oo t o, oo o. if [F(x) } (/x) is indeterminate in any of these forms for x = a, 

 its true value may be found by finding the true value of the logarithm of the 

 given expression. 

 Example: 



jxtan 



y\ logy= -tan*- 



