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278 CHAPTER 4 | Applications of Derivatives SOLUTION Because k—0and 1 — ¢™* — 0, we can apply I'Hospital's Rule: , A 1 —e* . et lim C = lim r » == rlim — k—0 k—0 k -0 | =p-]=p Thus the upper bound for the concentration is r, the same as the injection rate. U BT | Find lim ———— = | —cosx SOLUTION If we blindly attempted to use I'Hospital's Rule, we would get sin x _ Cos X lim ———= lim — = —o¢ t—=x- | —cosXx x—= sinx This is wrong! Although the numerator sin x — 0 as x — 7, notice that the denomi- nator (1 — cos x) does not approach 0, so I'Hospital's Rule can't be applied here. The required limit is, in fact, easy to find because the function is continuous at 7 and the denominator is nonzero there: _ sin x sin 7 0 lim — — =0 B v—=a- | — cos x l —cosm 1 —-(-1) Example 5 shows what can go wrong if you use I'Hospital's Rule without thinking. Other limits can be found using I'Hospital's Rule but are more easily found by other methods. (See Examples 2.4.3, 24.5, and 2.2.5 and the discussion at the beginning of this section.) So when evaluating any limit, you should consider other methods before using I'Hospital's Rule. B Which Functions Grow Fastest? L'Hospital's Rule enables us to compare the rates of growth of functions. Suppose we have two functions f(x) and g(x) that both become large as x becomes large, that is, lim f(x) = = and lim g(x) = = —-x We say that f(x) approaches infinity more quickly than g(x) if For example, we used I'Hospital's Rule in Example 2 to show that . e lim — = = r—em X and so the exponential function y = ¢* grows more quickly than y = x°, In fact y = ¢* Copyragiet 2006 Corgonpe Lcarmrg AL ghts Rowrved May sot be copeed scamsed, or deplicated 12 whole or in part D © clectromse nghts some thard pasty comteont may be suppressed from the cBook andor o hapraria) reveew bas decrned that amy smppremed costent doos et masar wily affect the overall keamusg cxpenerce (angage | crming reserves the rght o ramenve akdewnsl cormere 3t any timae f wubsagectt nghts roTKDoER fogure 1t