ࡱ> oqnWGq` -5bjbjqPqP ::- hhhhhhh4`8`8`8h8:^$;bLLLLQQQ]]]]]]]$_hxa]hQP0QQQ]hhLL]WWWQhLhL]WQ]WW]ZhhZL; P`8RuZ]]0^}Z,*bV*bZZ*bhZQQWQQQQQ]]WRQQQ^QQQQ3`8`8hhhhhh Limits Indeterminate Forms and LHopitals Rule I. Indeterminate Form of the Type  EMBED Equation.3  We have previously studied limits with the indeterminate form  EMBED Equation.3  as shown in the following examples:  Example 1:  EMBED Equation.3  Example 2:  EMBED Equation.3   EMBED Equation.3  [Note: We use the given limit  EMBED Equation.3 .] Example 3:  EMBED Equation.3 . [Note: We use the definition of the derivative  EMBED Equation.3  where  EMBED Equation.3  and a = 8.] Example 4:  EMBED Equation.3 . [Note: We use the definition of the derivative  EMBED Equation.3  where  EMBED Equation.3  and  EMBED Equation.3 .] 1 However, there is a general, systematic method for determining limits with the indeterminate form  EMBED Equation.3 . Suppose that f and g are differentiable functions at x = a and that  EMBED Equation.3  is an indeterminate form of the type  EMBED Equation.3 ; that is,  EMBED Equation.3  and  EMBED Equation.3 . Since f and g are differentiable functions at x = a, then f and g are continuous at x = a; that is,  EMBED Equation.3 = 0 and  EMBED Equation.3 = 0. Furthermore, since f and g are differentiable functions at x = a, then  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3 . Thus, if  EMBED Equation.3 , then  EMBED Equation.3  if EMBED Equation.3  and  EMBED Equation.3  are continuous at x = a. This illustrates a special case of the technique known as LHopitals Rule.  LHopitals Rule for Form  EMBED Equation.3  Suppose that f and g are differentiable functions on an open interval containing x = a, except possibly at x = a, and that  EMBED Equation.3  and  EMBED Equation.3 . If  EMBED Equation.3  has a finite limit, or if this limit is  EMBED Equation.3  or  EMBED Equation.3 , then  EMBED Equation.3 . Moreover, this statement is also true in the case of a limit as  EMBED Equation.3 or as  EMBED Equation.3   In the following examples, we will use the following three-step process: Step 1. Check that the limit of  EMBED Equation.3  is an indeterminate form of type  EMBED Equation.3 . If it is not, then LHopitals Rule cannot be used. Step 2. Differentiate f and g separately. [Note: Do not differentiate  EMBED Equation.3  using the quotient rule!] 2 Step 3. Find the limit of  EMBED Equation.3 . If this limit is finite,  EMBED Equation.3 , or  EMBED Equation.3 , then it is equal to the limit of  EMBED Equation.3 . If the limit is an indeterminate form of type  EMBED Equation.3 , then simplify  EMBED Equation.3  algebraically and apply LHopitals Rule again. Example 1:  EMBED Equation.3  Example 2:  EMBED Equation.3  Example 3:  EMBED Equation.3  Example 4:  EMBED Equation.3  Example 5:  EMBED Equation.3  [Use LHopitals Rule twice.] Example 6:  EMBED Equation.3 , or  EMBED Equation.3  where  EMBED Equation.3 . Example 7:  EMBED Equation.3 [This limit is not an indeterminate form of the type  EMBED Equation.3 , so LHopitals Rule cannot be used.] 3 II. Indeterminate Form of the Type  EMBED Equation.3  We have previously studied limits with the indeterminate form  EMBED Equation.3  as shown in the following examples: Example 1:  EMBED Equation.3   EMBED Equation.3  Example 2:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Example 3:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  limit does not exist. Example 4:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  because x < 0 and thus  EMBED Equation.3 ) =  EMBED Equation.3  4  EMBED Equation.3 . However, we could use another version of LHopitals Rule.  LHopitals Rule for Form  EMBED Equation.3  Suppose that f and g are differentiable functions on an open interval containing x = a, except possibly at x = a, and that  EMBED Equation.3  and  EMBED Equation.3 . If  EMBED Equation.3  has a finite limit, or if this limit is  EMBED Equation.3  or  EMBED Equation.3 , then  EMBED Equation.3 . Moreover, this statement is also true in the case of a limit as  EMBED Equation.3 or as  EMBED Equation.3   Example 1:  EMBED Equation.3  Example 2:  EMBED Equation.3  Example 3:  EMBED Equation.3  Example 4:  EMBED Equation.3  LHopitals Rule does not help in this situation. We would find the limit as we did previously. 5 Example 5:  EMBED Equation.3 =  EMBED Equation.3  Example 6:  EMBED Equation.3  Example 7:  EMBED Equation.3  [This limit is not an indeterminate form of the type  EMBED Equation.3 , so LHopitals Rule cannot be used.] III. Indeterminate Form of the Type  EMBED Equation.3  Indeterminate forms of the type  EMBED Equation.3  can sometimes be evaluated by rewriting the product as a quotient, and then applying LHopitals Rule for the indeterminate forms of type  EMBED Equation.3  or  EMBED Equation.3 . Example 1:  EMBED Equation.3  Example 2:  EMBED Equation.3   EMBED Equation.3  Example 3:  EMBED Equation.3  [Let  EMBED Equation.3 .] 6 IV. Indeterminate Form of the Type  EMBED Equation.3  A limit problem that leads to one of the expressions  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  is called an indeterminate form of type  EMBED Equation.3 . Such limits are indeterminate because the two terms exert conflicting influences on the expression; one pushes it in the positive direction and the other pushes it in the negative direction. However, limits problems that lead to one the expressions  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  are not indeterminate, since the two terms work together (the first two produce a limit of  EMBED Equation.3  and the last two produce a limit of  EMBED Equation.3 ). Indeterminate forms of the type  EMBED Equation.3  can sometimes be evaluated by combining the terms and manipulating the result to produce an indeterminate form of type  EMBED Equation.3  or  EMBED Equation.3 . Example 1:  EMBED Equation.3   EMBED Equation.3  Example 2:  EMBED Equation.3   EMBED Equation.3  V. Indeterminate Forms of the Types  EMBED Equation.3  Limits of the form  EMBED Equation.3   EMBED Equation.3  frequently give rise to indeterminate forms of the types  EMBED Equation.3 . These indeterminate forms can sometimes be evaluated as follows:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  7 The limit on the righthand side of the equation will usually be an indeterminate limit of the type  EMBED Equation.3 . Evaluate this limit using the technique previously described. Assume that  EMBED Equation.3 = L. (4) Finally,  EMBED Equation.3 . Example 1: Find  EMBED Equation.3 . This is an indeterminate form of the type  EMBED Equation.3 . Let  EMBED Equation.3   EMBED Equation.3 .  EMBED Equation.3 0. Thus,  EMBED Equation.3 . Example 2: Find  EMBED Equation.3 . This is an indeterminate form of the type  EMBED Equation.3 . Let  EMBED Equation.3   EMBED Equation.3 .  EMBED Equation.3 =  EMBED Equation.3 . Thus,  EMBED Equation.3 =  EMBED Equation.3 . Example 3: Find  EMBED Equation.3 . This is an indeterminate form of the type  EMBED Equation.3 . Let  EMBED Equation.3   EMBED Equation.3 .  EMBED Equation.3   EMBED Equation.3 . Thus,  EMBED Equation.3  =  EMBED Equation.3 . 8 Practice Sheet for LHopitals Rule (1)  EMBED Equation.3  (2)  EMBED Equation.3  (3)  EMBED Equation.3  (4)  EMBED Equation.3  (5)  EMBED Equation.3  (6)  EMBED Equation.3  (7)  EMBED Equation.3  (8)  EMBED Equation.3  (9)  EMBED Equation.3  (10)  EMBED Equation.3  (11)  EMBED Equation.3  9 (12)  EMBED Equation.3  (13)  EMBED Equation.3  (14)  EMBED Equation.3  (15)  EMBED Equation.3  (16)  EMBED Equation.3  (17)  EMBED Equation.3  (18)  EMBED Equation.3  (19)  EMBED Equation.3  (20)  EMBED Equation.3  (21) EMBED Equation.3  (22)  EMBED Equation.3  (23)  EMBED Equation.3  10 (24)  EMBED Equation.3  (25)  EMBED Equation.3  (26)  EMBED Equation.3  Solution Key for LHopitals Rule (1)  EMBED Equation.3  (2)  EMBED Equation.3   EMBED Equation.3  (3)  EMBED Equation.3   EMBED Equation.3  (4) Let  EMBED Equation.3  EMBED Equation.3 . Now, let  EMBED Equation.3   EMBED Equation.3 . Thus,  EMBED Equation.3  EMBED Equation.3  =  EMBED Equation.3 . 11 (5) Let  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3 . (6)  EMBED Equation.3   EMBED Equation.3 . (7) Let  EMBED Equation.3   EMBED Equation.3 . Thus,  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3 . (8)  EMBED Equation.3   EMBED Equation.3  limit does not exist. (9)  EMBED Equation.3   EMBED Equation.3 . (10)  EMBED Equation.3  12 (11) Let  EMBED Equation.3   EMBED Equation.3 . Thus,  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3 . (12)  EMBED Equation.3   EMBED Equation.3 . (13)  EMBED Equation.3   EMBED Equation.3 . (14)  EMBED Equation.3  (15)  EMBED Equation.3  (16)  EMBED Equation.3 . (17) Let  EMBED Equation.3  EMBED Equation.3 . Next, let  EMBED Equation.3   EMBED Equation.3 . Thus, 13  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3 . EMBED Equation.3  (18)  EMBED Equation.3 . (19)  EMBED Equation.3 . (20) Let  EMBED Equation.3   EMBED Equation.3 . Thus  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3 . (21)  EMBED Equation.3 . (22)  EMBED Equation.3 . (23)  EMBED Equation.3 . (24) Let  EMBED Equation.3  EMBED Equation.3  Let  EMBED Equation.3  14  EMBED Equation.3  Thus,  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  (25) Let  EMBED Equation.3   EMBED Equation.3 . Thus,  EMBED Equation.3   EMBED Equation.3  EMBED Equation.3 . (26)  EMBED Equation.3   EMBED Equation.3 . 15     #*1234:YZmnopq         - . 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CompObjlfObjInfonEquation Native o_1103537598F)Y,lim 3xx 2 "1x 2 x 2 x 2 +1x 2 = FMicrosoft Equation 3.0 DS Equation Equation.39qII + x!"" Ole sCompObjtfObjInfovEquation Native wlim 3x"1x 2 1+1x 2 =0"01+0=01=0 FMicrosoft Equation 3.0 DS Equation Equation.39qxDIlI + x!" l_1103537754  FY,.Ole {CompObj  |fObjInfo ~Equation Native _1103537857F..Ole CompObjfim 3x 3 "42x 2 +1= FMicrosoft Equation 3.0 DS Equation Equation.39qlII + x!" lim 3x 3 x 3 ObjInfoEquation Native _1103537998F..Ole "4x 3 2x 2 x 3 +1x 3 = FMicrosoft Equation 3.0 DS Equation Equation.39quII + x!" lim 3"4x 3 2CompObjfObjInfoEquation Native _1103613712F.1x+1x 3 =3"00+0=30!II FMicrosoft Equation 3.0 DS Equation Equation.39qltDIlI + x!"" lim  4x 2 +1  xOle CompObjfObjInfoEquation Native +1= FMicrosoft Equation 3.0 DS Equation Equation.39qllII + x!"" lim  4x 2 +1  xx+1x=_1103613771&F11Ole CompObjfObjInfo Equation Native _1103613962#F13Ole CompObj"$f FMicrosoft Equation 3.0 DS Equation Equation.39qluII + x!"" lim  4x 2 +1  " x 2 x+1xObjInfo%Equation Native _1103613995!0(F33Ole CompObj')fObjInfo*Equation Native P_1103614105-F35 FMicrosoft Equation 3.0 DS Equation Equation.39ql4LII ( x 2 ="x FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObj,.fObjInfo/Equation Native Ll0mIyI x=" x 2 FMicrosoft Equation 3.0 DS Equation Equation.39qlII + x!"" _1103614187+52F55Ole CompObj13fObjInfo4Equation Native _11036143077F55Ole CompObj68flim " 4x 2 +1x 2 x+1x=  FMicrosoft Equation 3.0 DS Equation Equation.39qlIԹI + x!"" lim " 4+1x 2ObjInfo9Equation Native _1140857274<F@f^@f^Ole  1+1x 2 =" 4  1="2 FMicrosoft Equation 3.0 DS Equation Equation.39q/ ""CompObj;=fObjInfo>Equation Native 6_1140857302SAF@8@8Ole CompObj@BfObjInfoCEquation Native g FMicrosoft Equation 3.0 DS Equation Equation.39qK?; + x!a lim f(x)=" FMicrosoft Equation 3.0 DS Equation Equation.39q_1140857312FF@8@8Ole CompObjEGfObjInfoHKw@ + x!a lim g(x)=" FMicrosoft Equation 3.0 DS Equation Equation.39q?H3 + x!+" Equation Native g_1140870210DNKFl?l?Ole CompObjJLfObjInfoMEquation Native [_1140870324PFl?l?Ole lim 3x 2 +5x"72x 2 "3x+1=+ x!+" lim 6x+54x"3=+ x!+" lim 64=32 FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjOQfObjInfoREquation Native J_1140870716I]UFl?A.7* + x!"" lim 3x"1x 2 +1=+ x!"" lim 32x=32+ x!"" lim 1x=32(0)=0Ole CompObjTVfObjInfoWEquation Native 4 FMicrosoft Equation 3.0 DS Equation Equation.39qO + x!" lim 3x 3 "42x 2 +1=+ x!" lim 9x 2 4x=+ x!" lim 18x4="_1140871176ZFAAOle CompObjY[fObjInfo\ FMicrosoft Equation 3.0 DS Equation Equation.39qcX` + x!"" lim  4x 2 +1  x+1=+ x!"" lim 8x2 4x 2 +1 Equation Native _1140935805Xb_FA'DOle CompObj^`f 1=+ x!"" lim 4x 4x 2 +1  ! 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FMicrosoft Equation 3.0 DS Equation Equation.39qPm + x!0 lEquation Native ">_1161668914QygF00:Ole #CompObjfh$fObjInfoi&Equation Native '_1161668915lF0:0:Ole /im 9" 81"5x  x=+ x!0 lim 9" 81"5x  x"9+ 81"5x  9+ 81"5x  =+ x!0 lim 81"(81"5x)x9+ 81"5x  ()=CompObjkm0fObjInfon2Equation Native 3_1161668916jtqF0:0 FMicrosoft Equation 3.0 DS Equation Equation.39qsI + x!0 lim 5xx9+ 81"5x  ()=+ x!0 lim 59+ 81"5x  =518Ole 8CompObjpr9fObjInfos;Equation Native < FMicrosoft Equation 3.0 DS Equation Equation.39qe0Zm + x!+" lim lnx 3 +2()ln5x 3 "1()=+ x!+" lim 3x 2 x 3 +215x 2 5x 3 "1=+ x!+" lim 3(5x 3 "1)15(x 3 +2)=+ x!+" lim 15x 3 "315x 3 +30=1_1161668917vF0 Ole GCompObjuwHfObjInfoxJ FMicrosoft Equation 3.0 DS Equation Equation.39qN*Z y=e x +1() ")2x !lny=lne x +1() ")2x ="2ln(e x +1)x!+ x!+Equation Native Kj_1161668918o{F Ole QCompObjz|Rf     /! 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FMicrosoft Equation 3.0 DS Equation Equation.39qD + x!+" _1161668921F!!Ole kCompObjlfObjInfonEquation Native o_1161668922=F!!Ole sCompObjtflim e x +1() )"2x =+ x!+" lim y=e "2 FMicrosoft Equation 3.0 DS Equation Equation.39q + x!0 lObjInfovEquation Native w_1161668923FOle ~im sin(4x)"2sin(2x)x 3 =+ x!0 lim 4cos(4x)"4cos(2x)3x 2 =+ x!0 lim "16sin(4x)+8sin(2x)6x= FMicrosoft Equation 3.0 DS EqCompObjfObjInfoEquation Native _1161668924Fuation Equation.39q,t + x!0 lim "64cos(4x)+16cos(2x)6="486="8 FMicrosoft Equation 3.0 DS EqOle CompObjfObjInfoEquation Native yuation Equation.39q](,\ + x!0 lim 1sinx"1x[]=+ x!0 lim x"sinxxsinx()=+ x!0 lim 1"cosxxcosx+sinx=_1161668925FOle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q9. + x!0 lim sinx"xsinx+cosx+cosx=02=0Equation Native _1161668926FOle CompObjf FMicrosoft Equation 3.0 DS Equation Equation.39q:m + x!+" lim xe )1x "1()=+ x!+" lim e )1x "1)1xObjInfoEquation Native *_1161668927FMOle =+ x!+" lim e )1x )"1x 2 ())"1x 2 =+ x!+" lim e )1x =e 0 =1. FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjfObjInfoEquation Native n_1161668928FMMR(Z + x!0 +  lim  x 3 lnx=+ x!0 +  lim lnxx ")13 =+ x!0 +  lim )1x")13x ")43 =+ x!0 +  lim "3x )43 x=+ x!0 +  lim ("3 x 3 )=0. FMicrosoft Equation 3.0 DS Equation Equation.39q=Z + x!0 lOle CompObjfObjInfoEquation Native im ln2x+15x+1()x=+ x!0 lim 5x+12x+1()2(5x+1)"5(2x+1)(5x+1) 2 ()1=+ x!0 lim "3(2x+1)(5x+1)="3_1161668929FMOle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q&QZ y=1x! FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native B_1161668930FOle CompObjfObjInfoEquation Native _1161668931FOle (^ + x!+" lim 1+ex() )x2 =+ y!0 +  lim 1+ey() )12y FMicrosoft Equation 3.0 DS EqCompObjfObjInfoEquation Native _1161668932Fuation Equation.39qk0}` z=1+ey() )12y !lnz= FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native pSZ ln1+ey() )12y =ln(1+ey)2y!+ y!0 +  lim lnz=+ y!0 +  lim ln(1+ey)2y=+ y!0 +  lim e1+ey2=e2 FMicrosoft Equation 3.0 DS Equation Equation.39q1w4" + y!0 +  lim lnz=e2!ln+ y!0 +  lim _1161668933FOle CompObjfObjInfoEquation Native M_1161668934FxOle CompObjfz()=e2!+ y!0 +  lim z=e )e2 ! 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