Schaum's Outline of Theory and Problems of Beginning Calculus

CHAP. 361 L'HOPITAL'S RULE; EXPONENTIAL GROWTH AND DECAY 285This example illustrates the importance of having the function expressed as a fraction "in the right way."e-xSuppose we had chosen "the wrong way," and tried to evaluate the same limit as lirn F. Then repeatedapplication of L'Hbpital's rule would have givenand we should never have arrived at a definite value.(e) See Problem 34.13(c). By L'H8pital's rule,e-x (- l)e-. (- 1)2e-xlim 7 = lim =(-n)x-'"+')limX(- 1)2(n)(n + 1 )~-("+~) -X++m- ...FURTHER EXAMPLES(1) Find lirn (In x)lIX.X++O3Since In x --* + 00 and l/x 40, it is not clear what the limit is. Let y = (In x)'/~. Then In y =1- In (In x). Hence, by L'HGpital's rule,XTherefore,(2) Find lirn x'/~.X++W-.-1 1In (In x) lnx x 1lim In y = lim - lim -- - lim -=0X++m x++m x X++a) 1 ,++,xInx1Since x -P +00 and l/x +O, the limit is not obvious. Let y = x1IX. Then In y = - In x, andXIn xlim In y = lim - - 0 by (e) above. Hence, lirn y = lim e'" = e0 = 1.Warning:results.When the conditions for L'HBpital's rule do not hold, use of the rule usually leads to falsex2+1 22+1 5EXAMPLElim - -=x+2x2- 1 22- 1 3If we used L'Hbpital's rule, we would conclude mistakenly that2xlim x2 + - lim - = lim 1 = 1x+2 x2 - 1 x+2 2x x-+236.2 EXPONENTIAL GROWTH AND DECAYExample (d) above shows that ex grows much faster than any power of x. There are many naturalprocesses, such as bacterial growth or radioactive decay, in which quantities increase or decrease at an"exponential rate."