Indeterminate Forms and LHospzitals Rule

8/21/2019 Indeterminate Forms and LHospzitals Rule

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Section 3.7 Indeterminate Forms and LHospitals Rule 2010 Kiryl Tsiscan!a

Indeterminate Forms and LHospitals Rule

TH"#R"$ %LHospitals Rule&' Suppose ( and ) are di erentia*le and ) %+& , 0 near a

%e+cept possi*ly at a&. Suppose tat

lim ( %+& , 0 and lim )%+& , 0+ a + a

or tat

lim ( %+& , and lim )%+& ,+ a + a

%In oter -ords -e a/e an indeterminate (orm o( type 0

0 or .& Ten

( %+& ( %+&lim

)%+& , lim ) %+&+ a + a

i( te limit on te ri)t side e+ists %or is or &.

Indeterminate Forms o( Type 00 and

"$L"S'

+ 21. Find lim

7+ 4 3 .+

Solution 1' 5e a/e

+ 2 + 2 + 2 2lim , lim , lim , 0+ + + +

7+ 4 3 , , lim 4 7 4 7 4 0 , 77+43 7+ 3 3+ + + ++ + + +

Solution 2' 5e a/e

+ 2 %+ 2& %+& 2 + 2lim

7+ 4 3 , , lim %7+ 4 3& , lim %7+& 4 3 , lim 7+ 4 3+ + + +

1 0 , lim

7 1 4 0 , lim7 , 7+ +

In sort+ 2 %+ 2&

lim7+ 4 3 , lim %7+ 4 3& , lim 7 , 7+ + +

+ 4 + 4 + 4 + 4 + 4 1 6 3 22. Find lim 2+ 4 + 4 + 4 + 4 + 4 1 . 6 3 2+ 1


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+ 4 + 4 + 4 + 4 + 4 1 6 3 22. Find lim

2+ 4 + 4 + 4 + 4 + 4 1 . 6 3 2+

Solution 1' 5e a/e

+ 4 + 4 + 4 + 4 + 4 1 + 4+ 6 4+ 3 4+ 2 4+41 6 3 2

lim +2+ 4 + 4 + 4 + 4 + 4 1 , , lim 6 3 2 2+ 4+ 4+ 4+ 4+41 6 3 2+ ++

4 4 4 4 4 1 4 4 4 4 4+ + 6 +3 +2 + 1 1 1 1 1 1, lim , lim+ + + + + + + + + + + 2 3 6

2 4 4 4 4 44 4 4 4 4 1 1 1 1 12+ + + + + 1 6 3 2+ ++ + + + ++ + + + + + 2 3 6

, 1 4 0 4 0 4 0 4 0 4 02 4 0 4 0 4 0 4 0 4 0 , 1 2

Solution 2' 5e a/e

+ 4 + 4 + 4 + 4 + 4 1 %+ 4 + 4 + 4 +4 + 4 1& 6 3 2 6 3 2lim

2+ 4 + 4 + 4 + 4 + 4 1 , , lim %2+ 4 + 4 + 4 +4 + 4 1& 6 3 2 6 3 2+ +

+ 4 6+ 4 3+ 4 2+ 4 1 %+ 4 6+ 4 3+4 2+ 4 1&6 3 2 6 3 2, lim

10+ 4 6+ 4 3+ 4 2+ 4 1 , , lim %10+ 4 6+ 4 3+4 2+ 4 1&6 3 2 6 3 2+ +

20+ 4 12+ 4 + 4 2 %20+ 4 12+ 4 + 4 2&3 2 3 2, lim

60+ 4 12+ 4 + 4 2 , , lim %60+ 4 12+ 4 + 4 2&3 2 3 2+ +

0+ 4 26+ 4 %0+ 4 26+ 4 &2 2, lim

120+ 4 26+ 4 , , lim %120+ 4 26+ 4 &2 2+ +

120+ 4 26 %120+ 4 26& 120, lim

260+ 4 26 , , lim %260+ 4 26& , lim 260 , 120260 , 1 2+ + +

+ 4 23. Find lim

ln%+ 4 3& .+ 2

Solution' 5e a/e

+ 4 2 %+ 4 2& %+ 4 2& + 4 2lim , lim

1 1ln%+ 4 3& , 0 0 , lim %ln%+ 4 3&& , lim+ 2 + 2 + 2 + 2

+ 4 3 %+ 4 3& + 4 3 %+ 4 3 &

1 4 0 1 1 %+ 4 3& + 4 3

, lim , lim , lim , lim %+ 4 3&1 1 1 1 , lim+ 2 + 2 + 2 + 2 + 2

+ 4 3 %1 4 0& + 4 3 + 4 3 %+ 4 3&

, 2 4 3 , 1

In sort+ 4 2 %+ 4 2& 1lim , lim%+ 4 3& , 1ln%+ 4 3& , lim %ln%+ 4 3&& , lim %+ 4 3& 1+ 2 + 2 + 2 + 23+6. Find lim + 4 + 1 .2+ 2


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3+6. Find lim

+ 4 + 1 .2+

Solution' 5e a/e

3 %3 & 3 ln 3 %3ln 3&+ + + +lim

+ 4 + 1 , , lim %+ 4 + 1& , lim 2+ 4 1 , , lim %2+ 4 1&2 2+ + + +

ln 3 %3 & ln 3 3 ln 3+ +, lim

2 , lim 2 , %8.9.".&+ +

ln +. Find lim

+ .+

Solution' 5e a/e

ln + %ln +& + + + 11 1

lim , lim , 0+ , , lim %+ & , lim + + + , lim +1 1 11:2 1:2 1:2 1:2+ + + + +

2 2 2

+ tan +. Find lim .

+3+ 0

Solution 1' 5e a/e

+ tan + %+ tan +& sec + %+& sec+ 2 2lim , 0 , lim

+ 0 , lim %+ & , lim 3+ 3+3 3 2 2+ 0 + 0 + 0 + 0

%1 sec +& 1 sec+2 2, lim ,

3+ 3 lim +2 2+ 0 + 0

1 sec +2Since lim is an indeterminate (orm o( type 0

+ 0 -e can use LHospitals Rule a)ain.2+ 0;ut it is easier to do tri)onometry instead. 9ote tat 1 sec + , tan+. Tere(ore2 2

1 sec + tan + tan + sin 2 + sin2+2 2 2, , , , cos + 12

3 lim + 3 lim + 3 lim + 3 lim + 3 lim+2 2 2 2 2+ 0 + 0 + 0 + 0 + 0

sin + sin + 2 2 sin +22, , 0 , sin + ,

3 lim + 0 , 3 lim + 3 lim + 3 lim+2+ 0 + 0 + 0 + 0

sin + 2 sin u, , lim 2 ,

3 lim + u , 1 , 3 % 1& 3 2 , 12 3+ 0 u 0

In sort

+ tan + %+ tan +& 1 sec + tan +sin+2 2 2lim , lim , ,

+ %+ & , 3 lim + 3 lim + 3 lim+3 3 2 2 2+ 0 + 0 + 0 + 0 + 0

2

, sin + , , 1223 lim + 3 % 1& 3+ 0

Solution 2


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+ 4 sin +7. Find lim

+ 4 cos + .+

Solution 1' 5e a/e

4 1 4+ 4 sin + +4sin + + sin + sin +lim , lim , lim+ + + +

+ 4 cos + , , lim 4 1 4+4cos + + cos + cos ++ + + ++ + + +

sin + cos +It is easy to so- tat lim

+ , 0 and lim + , 0 *y te S?uee@e Teorem. Tere(ore+ +

1 4+ 4 sin + sin +lim , 1 4 0+

+ 4 cos + , lim 1 4 1 4 0 , 1cos ++ ++

Solution 2%


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Indeterminate Forms o( Type and 0

"$L"S'

10. Find lim %+ ln +&.+

Solution 1' 5e a/e

lim %+ ln +& , D E , lim + 1 + ln + + 1 ln ++ , lim ++ + +

9ote tatln + %ln +& + 11

lim+ , , lim + , lim 1 , lim + , 0+ + + +

tere(ore

lim %+ ln +& , lim + 1 ln + + %1 0& , %8.9.".&+ , lim+ + +

Solution 2' 5e a/e

+

lim %+ ln +& , D E , lim %ln%e & ln +& , lim ln e+++ + +

e %e & e+ + +, ln lim

+ , , ln lim + , ln lim 1 , %8.9.".&+ + +

1 111. Find lim

ln + + 1 .+ 1


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1 111. Find lim

ln + + 1 .+ 1

Solution' 5e a/e

1 1 1 %+ 1& ln + 1limln + + 1 , D E , lim ln + %+ 1& ln + %+ 1&+ 1 + 1

+ 1 ln + %+ 1 ln +&, lim

ln +%+ 1& , 0 0 , lim %ln +%+ 1&&+ 1 + 1

1 1+ 1 %ln +& +, lim+ 1%ln +& %+ 1& 4 ln + %+ 1& , lim+ 1 + 1

+ 4 ln +

1 1 1 + 1+ + + +

, lim , lim+ 1+ 1+ 1 + 1

+ + 4 ln + ++ 4 ln + +

+ 1 %+ 1& + 1, lim

+ 1 4 + ln + , 0 0 , lim %+ 1 4 + ln +& , lim + 1 4 + ln + 4 +%ln +&+ 1 + 1 + 1

1 0 1, lim , lim2 4 ln + , 1 2 4 0 , 1 2+ 1 + 1

1 0 4 1 ln + 4 + 1+

In sort

1 11 1 + 1 ln + %+ 1 ln +& +

lim+ 1ln + + 1 , lim ln + %+ 1& , lim %ln + %+ 1&& , lim+ 1 + 1 + 1 + 1

+ 4 ln +

+ 1 %+ 1& 1, lim

+ 1 4 + ln + , lim %+ 1 4 + ln +& , lim2 4 ln + , 12+ 1 + 1 + 1

12. Find lim %sin + ln +&.+ 0 4


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12. Find lim %sin + ln +&.+ 0 4

Solution 1' 5e a/e

sin + ln + %sin +& sin + ln + ln +1lim %sin + ln +& , D0 E , lim ,

1 , lim %sin +& 1 , lim %sin +&1 1+ 0 + 0 + 0 + 04 4 4 4

%ln +& + +1 1, lim

%%sin +& & , lim %sin +& %sin +& , lim%sin +&cos +2 21+ 0 + 0 + 04 4 4

+ + sin + sin +1 2 2, lim

%sin +& cos + + sin + , lim + cos +22+ 0 + 04 4

5e can no- proceed in t-o di erent -ays. "iter

sin + sin + sin +2lim

+ cos + , 0 0 , lim + cos ++ 0 + 04 4

sin + sin +, lim

+ lim cos + , 1 sin 0 cos 0 , 1 01 , 0+ 0 + 04 4or

sin + %sin +& 2 sin + cos +2 2lim

+ cos + , 0 0 , lim %+ cos +& , lim + cos + 4 +%cos +&+ 0 + 0 + 0

4 4 4

2 sin + cos +, lim

cos + + sin + , 2 sin 0 cos 0cos 0 0 sin 0 , 2 0 11 0 0 , 0+ 0 4

In sort

ln + %ln +& + 1lim %sin + ln +& , lim , lim

%sin +& %%sin +& & , lim %sin +&cos +1 21+ 0 + 0 + 0 + 04 4 4 4

sin + sin + sin +2

, lim+ cos + , lim + cos + , 1 0 , 0+ 0 + 04 4

Solution 2' 5e a/e

sin + ln + sin + ln + ln + sin +1lim %sin + ln +& , D0 E , lim

1 , lim 01 ln + , lim ln+ , 01 1+ 0 + 0 + 0 + 04 4 4 4

%sin +& cos + cos + ln+ cos +2

, lim , limln + %ln +& , lim ln + + ln+ ln+ +ln + , lim 2 2 221 1 1+ 0 + 0 + 0 + 04 4 4 4

ln + cos + ln + ln + ln2+2 2 2, lim , lim lim cos + , lim , , lim

+ + + % +&1 1 1 1+ 0 + 0 + 0 + 0 + 04 4 4 4 4

2 ln + %ln +& 2 ln + + 2 ln + + + 2 ln +1 1, lim , lim , lim ,+ + + + , lim +2 2 2 1+ 0 + 0 + 0 + 04 4 4 4%2 ln +& 2+ 2+ + 2+1 1 2, lim , lim , lim+ + + 1 , 2 0 1 , 0%+ & , lim 2 2 21+ 0 4 + 0 4 + 0 4 + 0 47


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Indeterminate Forms o( Type 0 and 10 0

13. Find lim + .1:++

Solution %/ersion 1&' 9ote tat lim + is type o( an indeterminate (orm. ut1:+ 0+

y , + 1:+

ten

ln y , ln + , 11:++ ln + , ln ++

5e a/e

ln + %ln +& + 11lim

+ , , lim + , lim 1 , lim + , 0+ + + +

Tere(ore

lim + , e , 11:+ 0+

Solution %/ersion 2&' 5e a/e

% ln +&ln + + 1 1lim lim lim limlim + , lim e , lim e

ln +

, e , e , e , e, e, 11:+ ln + 1 : + 0+ + 1 ++ + + +++ + +

16. Find lim %tan +& .2+ p

+ p:2

A


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16. Find lim %tan +& .2+ p+ p:2

Solution' 9ote tat lim %tan +& is type o( an indeterminate (orm. ut2+ p 0+ p:2

y , %tan +&2+ p

ten

ln y , ln %tan +& , %2+ p& ln%tan +&2+ p

%2+ p & ln%tan +&1, %2+ p& ln%tan +&

1 , %2+ p& %2+ p& 1 , ln%tan +&%2+ p&1 1

5e a/e

ln%tan +& Dln%tan +&E %tan +&1

lim , , lim tan +%2+ p& D%2+ p& E , lim % 1&%2+ p& %2+ p&1 21+ p:2 + p:2 + p:2

1 1 1sec +1 2 cos +s in + 2 s in + cos +2

, lim , 1tan + c os + co s +% 1&%2+ p& 2 , 1 2 lim %2+ p& 2 lim %2+ p&2 2 2+ p :2 + p :2 + p :2

sin + cos +%2+ p& %2+ p&1 1 2 2, 1 , 1 , 1sin + cos + sin + cos +

2 lim %2+ p& 2 lim %2+ p& sin + cos +%2+ p& 2 limsin + cos +2 2 2+ p:2 + p:2 + p:2

%2+ p& %2+ p& D%2+ p & E2 2 2, 1

2 lim 2 lim cos + , 0 0 , 1 2 lim %cos +&sin cos + , 1p+ p:2 + p:2 + p:22

2 2 p2%2+ p& %2+ p& 2%2+ p& 2 2 p 2, 12 lim sin + , 1 2 lim sin + , 1 2 sin p+ p:2 + p :2

2

2 0 2, 1

2 1 , 0Tere(ore

lim %tan +& , e , 12+ p 0

+ p:2

In sort

y , %tan +& , ln y , %2+ p& ln%tan +& , ln%tan +&2+ p%2+ p& 1

5e a/e

sec +ln%tan +& Dln%tan +&E1

%2+ p&2 2lim , lim tan +%2+ p& % 1&%2+ p& 2 , 1 2 limsin + cos +D%2+ p& E , lim1 21+ p:2 + p :2 + p :2 + p:2

%2+ p& D%2+ p&E2 2, 1

2 lim cos + , 0 0 , 1 2 lim%cos +&+ p:2 + p:2

2 2 p2%2+ p& 2 2 p 2, 1 , 02 lim sin + , 1 2 sin p+ p:2 2Tere(ore lim %tan +& , e , 1.2+ p 0+ p:2 B


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1. Find lim + .++ 0 4

Solution' 9ote tat lim + is 0 type o( an indeterminate (orm. ut+ 0

+ 0 4

y , ++

ten+ ln +1

ln y , ln + , + ln + , + ln ++1 , + + 1 , ln + +1 1

5e a/e

ln + %ln +& + 1lim , , lim

+ %+ & , lim +1 1 2+ 0 + 0 + 04 4 4

+ + +1 2, lim , lim

+ + 1 , 01 , 02 2+ 0 + 04 4

Tere(ore

lim + , e , 1+ 0

+ 0 4

In sort

y , + , ln y , + ln + , ln ++ + 1

5e a/eln + %ln +& + 1

lim , lim , lim + , 0+ %+ & , lim +1 1 2+ 0 + 0 + 0 + 04 4 4 4

Tere(ore

lim + , e , 1+ 0

+ 0 4

1. Find lim %tan +& .+

+ 0 4

10


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1. Find lim %tan +& .++ 0 4

Solution' 9ote tat lim %tan +& is 0 type o( an indeterminate (orm. ut+ 0

+ 0 4

y , %tan +&+

ten

+ ln%tan +&1ln y , ln%tan +& , + ln%tan +& , + ln%tan +&+

1 , + + 1 , ln%tan +&+1 1

5e a/e

%tan +&ln%tan +& %ln%tan +&&

1

lim , , lim tan ++ %+ & , lim +1 1 2+ 0 + 0 + 04 4 4

sec + %+& sec+ 1 2 1 2, lim , limtan + tan +

+ +2 2+ 0 + 04 4

sec + sec 0 1 12 2s in + s in +

, lim , limcos + co s 0+ +2 2+ 0 4 + 0 4

+ sin +1 1 2, lim , limsin + sin +

+ + + sin +2 2 2+ 0 + 04 4

+ %+ &2 2, lim

sin + , 0 0 , lim % sin +&+ 0 4 + 0 4

10+ 10+, lim

cos + %+& , lim cos + , 10 0cos 0 , 0+ 0 + 04 4

Tere(ore

lim %tan +& , e , 1+ 0

+ 0 4

In sort

y , %tan +& , ln y , + ln%tan +& , ln%tan +&++ 1

5e a/e

sec+ 1 2sec + ln%tan +& %ln%tan +&& 1 2 s in +lim , lim , limtan + c os +

+ %+ & , lim + +1 1 2 2+ 0 + 0 + 0 + 04 4 4 4

+ %+ & 10+1 2 2, lim , limsin +

+ sin + , lim % sin +& , limcos + , 02+ 0 + 0 + 0 + 04 4 4 4Tere(ore lim %tan +& , e , 1+ 0+ 0 4 11


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17. Find lim %sin 2+& .tan 3++ 0 4

Solution' 9ote tat lim %sin 2+& is 0 type o( an indeterminate (orm. uttan 3+ 0

+ 0 4

y , %sin 2+&tan 3+

ten

ln y , ln %sin 2+& , tan 3+ ln%sin 2+& , sin 3+ ln%sin 2+&tan 3+cos 3+

5e a/e

lim %sin 3+ ln%sin 2+&&sin 3+ ln%sin 2+&

lim + 04

%sin 3+ ln%sin 2+&& , D0 Ecos 3+ , lim cos 3+ , lim+ 0 4 + 0 4

+ 04

sin 3+ ln%sin 2+& %sin 3+& sin 3+ ln%sin 2+& ln%sin 2+&1, lim ,

1 , lim %sin 3+& 1 , lim %sin 3+&1 1+ 0 + 0 + 04 4 4

Dln%sin 2+&E sin 2+ %sin 2+& sin2+ cos 2+ %2+&1 1, lim

%sin 3+& %sin 3+& , lim %sin 3+&cos 3+ %3+&%%sin 3+& & , lim1 2 2+ 0 + 0 + 04 4 4

sin 2+ cos 2+ 2 sin 2+ cos 2+ sin2+1 1 1, lim lim

%sin 3+& cos 3+ 3 , 2 3 lim %sin 3+& cos 3+ , 2 3 lim%sin 3+&2 2 2+ 0 + 0 + 0 + 04 4 4 4

sin 2+ sin 2+ sin 3+ sin 3+ %sin3+&1 2 2 2, 2

3 lim 3 lim sin 2+ , 0 0 , 2 3 lim%sin 2+&%sin 3+& sin 2+ sin 3+ , 222+ 0 + 0 + 04 4 4

2 sin 3+ %sin 3+& 2 sin 3+ cos 3+ %3+&, 2

3 lim cos 2+ %2+& , 2 3 lim cos 2+ 2+ 0 + 04 4

2 sin 3+ cos 3+ 3 2 sin 0 cos 0 3 2 0 1 3, 2 3 lim cos 2+ 2 , 2 3 cos 0 2 , 2 3 1 2 , 0+ 0 4

Tere(ore

lim %sin 2+& , e , 1tan 3+ 0

+ 0 4

In sort

y , %sin 2+& , ln y , tan 3+ ln%sin 2+& , sin 3+ ln%sin 2+&tan 3+cos 3+

5e a/esin 3+ ln%sin 2+& ln%sin 2+& Dln%sin 2+&E

lim %sin 3+ ln%sin 2+&& , lim , limcos 3+ , lim %sin 3+& %%sin 3+&&1 1+ 0 + 0 + 0 + 04 4 4 4

sin 2+ cos 2+ 2 sin 2+ sin 3+ %sin3+&1 1 2 2, lim , 2

%sin 3+& cos 3+ 3 , 2 3 lim %sin 3+& 3 lim sin 2+ , 23 lim%sin 2+&2 2+ 0 + 0 + 0 + 04 4 4 42 sin 3+ cos 3+ 3 2 sin 0 cos 0 3, 2 3 lim cos2+ 2 , 2 3 cos 0 2 , 0+ 0 4Tere(ore lim %sin 2+& , e , 1.tan 3+ 0+ 0 4 12


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+ 4 1 +1A. Find lim .

+ 4 2+

+ 4 1 + +

Solution 1' 9ote tat lim is 1 type o( an indeterminate (orm. ut y , + 4 1+ 4 2 + 4 2+

ten

+ ln + 4 1 + + ln + 4 1ln + 4 11+ + 4 2 + 4 2+ 4 2

ln y , ln + 4 1 , + ln + 4 1+ 4 2 + 4 2 , 1 , + 1 , +1 1

5e a/e

1 + 4 1

ln + 4 1 ln + 4 1 + 4 2+ 4 2 + 4 2 +41lim , 0 +42

+ 0 , lim %+ & , lim +1 1 2+ + +

+ 4 2 %+ 4 1& %+ 4 2& %+ 4 1&%+ 4 2& + 4 2 1 %+ 4 2& 1 %+ 4 1&

+ 4 1 %+ 4 2& + 4 1 %+ 4 2&2 2, lim , lim

+ +2 2+ +

+ 4 2 + 4 2 + 1 + 4 2 1 1

+ 4 1 %+ 4 2& + 4 1 %+ 4 2& %+ 4 1&%+ 4 2&2 2

, lim , lim , lim+ + +2 2 2+ + +

1%+ 4 1&%+ 4 2&2

%+ 4 1&%+ 4 2& + + +2 2, lim

+ + %+ 4 1&%+ 4 2& , lim %+ 4 1&%+ 4 2& , , lim +4 3+ 4 12 2 2+ + +

+ +2 2

1+ +2 2, lim , lim , lim , 1

+ 4 3+ 4 1 + 1 4 0 4 0 , 12 2+ + +

1 4 34 3+ 4 1 + 4 1++ + + + 22 2 2 2

Tere(ore+ 4 1 +

lim , e 1+ 4 2+

In sort

ln + 4 1+ + 4 2

y , + 4 1 , ln y , + ln + 4 1

+ 4 2 + 4 2 , +1

5e a/e

+ 4 2 1ln + 4 1 ln + 4 1

+ 4 2 + 4 2 + 4 1 %+ 4 2& +22lim , lim , lim

+ %+ & , lim + %+ 4 1&%+ 4 2&1 1 2+ + + ++2, lim +4 3+ 4 1 , 12++ 4 1 +Tere(ore lim , e .1+ 4 2+ 13


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Solution 2' 5e a/e

+ 4 1 + + 4 2 1 +lim , lim

+ 4 2 + 4 2+ +

+

+ 4 2 1, lim+ 4 2 + 4 2+

+

, lim 1 1+ 4 2+

% + 2& ++ 2

, lim 1 4 1+ 2+

++ 2 + 2

, lim 1 4 1+ 2+

u + 2

Since lim 1 4 1 , e it (ollo-s tat lim 1 4 1 , e. Tere(oreu + 2u +

+

+ 4 1 + + 2 + 2

lim , lim 1 4 1+ 4 2 + 2+ +

, lim e+

+ 2

+

lim +, e + 2+

lim +

, e % + 2 &+

, e 1

C#$R"' 5e a/e

+ 4 1 +lim , 0

3+ 4 2+

since+

+ 4 1 1

lim , 03+ 4 2 , 1 3 and lim 3+ +

Similarly+

3+ 4 1lim , %8.9.".&

+ 4 2+

since 3+ 4 1lim 3 , %8.9.".&++ 4 2 , 3 and lim+ ++1B. Find lim 1 4 1 .++ 16


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+

1B. Find lim 1 4 1 .++

Solution' 5e a/e+

lim 1 4 1 , e++

*y de(inition o( e. 9ote tat -e cant use te approac descri*ed in

"+ample 13 since it is *ased +o( te (ormula %ln +& , 1 1 4 1, e

+ . ;ut tis result -as deduced (rom te (act tat lim ++%see ppendi+ &. Tis is Circular Reasonin) -ic is a lo)ical (allacy.

Ho-e/er one can applyLHospitals Rule to modi(ications o( tis limit.

20. Find lim %1 4 sin 7+& .cot +

+ 0 4

Solution' 9ote tat lim %1 4 sin 7+& is 1 type o( an indeterminate (orm. utcot+

+ 0 4

y , %1 4 sin 7+& cot +

ten

ln y , ln %1 4 sin 7+& , cot + ln%1 4 sin 7+& , cos + ln%1 4 sin 7+&cot +sin +

5e a/e

cos + ln%1 4 sin 7+& ln%1 4 sin 7+&lim cos + lim

sin + , lim sin ++ 0 + 0 + 04 4 4

%1 4 sin 7+&ln%1 4 sin 7+& Dln%1 4 sin 7+&E 1, lim 14sin 7+

sin + , 0 0 , lim %sin +& , lim cos + %+&+ 0 + 0 + 04 4 4

cos 7+ %7+& cos 7+ 7 1 71 1 1

, lim 14sin 7+ 14sin 7+ 140cos + , lim cos + , 1 , 7 + 0 4 + 0 4

Tere(ore

lim %1 4 sin 7+& , ecot + 7:

+ 0 4

In sort

y , %1 4 sin 7+& , ln y , cot + ln%1 4 sin 7+& , cos + ln%1 4 sin 7+&cot +

sin +

5e a/e

cos + ln%1 4 sin 7+& ln%1 4 sin 7+& Dln%1 4 sin 7+&Elim

sin + , lim sin + , lim %sin +&+ 0 + 0 + 04 4 4

cos 7+ 7 1 71 1, lim 14sin 7+ 140cos + , 1 , 7 + 0 4Tere(ore lim %1 4 sin 7+& , e .cot + 7:+ 0 4 1


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ppendi+

TH"#R"$' Te (unction ( %+& , sin + is di erentia*le and

( %+& , cos +

roo(' 5e a/e

sin%+ 4 & sin +%sin +& , lim

0

D5e use sin%a 4 & , sin a cos 4 cos a sin E

sin + cos 4 cos + sin sin + sin + cos sin +, lim

, lim 4 cos + sin 0 0

sin +%cos 1& cos 1 sin , lim

4 cos + sin , sin + lim 4 cos + lim 0 0 0

, sin + 0 4 cos + 1 , cos +

TH"#R"$' Te (unction ( %+& , lo) + is di erentia*le anda

( %+& , 1+ ln a

roo(' 5e a/e

d lo) %+ 4 & lo) + 1 + 4 +& , lim a a

d+ %lo) , lim lo) +a a 0 0

1 1 +, lim 1 4 1 4

lo) + , lim + lo) +a a 0 0

+: 1:%:+&

, 1 lo) 1 4 , 1 lo) 1 4 + lim a + + lim a + 0 0

1:%:+& ln e, 1 lim 1 4 , lim %1 4 u& 1:u , e , 1 e , 1

+ lo) a + + lo) a +ln a , 1+ ln a 0 u 0

1


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ppendi+ ;

sin +1. Find lim

2+ .+ 0

Solution 1' 5e a/e

sin + %sin +& cos +lim

2+ , 0 0 , lim %2+& , lim 2 , cos 0 2 , 1 2+ 0 + 0 + 0

Solution 2' 5e a/esin + sin +

lim2+ , 0 0 , 1 2 lim + , 1 2 1 , 1 2+ 0 + 0

sin 2+2. Find lim

+ .+ 0

Solution 1' 5e a/e

sin 2+ %sin 2+& 2 cos 2+lim

+ , 0 0 , lim + , lim 1 , 2 cos 0 1 , 2+ 0 + 0 + 0

Solution 2' 5e a/esin 2+ sin 2+

lim+ , 0 0 , 2 lim 2+ , 2 1 , 2+ 0 + 0

sin 2+3. Find lim

+ .+ 0

Solution 1' 5e a/e

sin 2+ %sin 2+& 2 cos 2+lim+ , 0 0 , lim %+& , lim , 2 cos 0 , 2 + 0 + 0 + 0

Solution 2' 5e a/esin 2+ sin 2+

lim+ , 0 0 , 2 lim 2+ , 2 1 , 2 + 0 + 0

sin 2+6. Find lim

sin + .+ 0

Solution 1' 5e a/e

sin 2+ %sin 2+& 2 cos 2+li

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Indeterminate Forms and LHospzitals Rule