# Determinate and Indeterminate Limit Forms

## Transcript Of Determinate and Indeterminate Limit Forms

Determinate and Indeterminate Limit Forms

Some limits can be determined by inspection just by looking at the form of the limit – these predictable limit forms are called determinate. Other limits can’t be determined just by looking at the form of the limit and can only be determined after additional work is done – these unpredictable limit forms are called indeterminate.

Determinate Limit Forms:

Assuming that the functions involved in the limit are defined:

1. The limit forms a , for any number a, and bounded result in a limit of 0.

2. The limit form results in a limit of .

3. The limit form results in a limit of 0.

4. The limit form a , for 1 a 1, results in a limit of 0.

5. The limit form a , for a 1, results in a limit of .

6. The limit form a , for a 0 , results in a limit of .

7. The limit form a , for a 0 , results in a limit of 0.

8. The limit forms , , and a , for a 0 , result in a limit of . 0 a

9. The limit forms , , and a , for a 0 , result in a limit of . 0 a

10. The limit forms and result in a limit of .

11. The limit forms , , and result in a limit of .

12. The limit forms a 0 , for any number a, and bounded 0 result in a limit of 0.

13. The limit forms a , for any number a, and bounded result in a limit of .

14. The limit form a0 , for a 0, results in a limit of 1.

Indeterminate Limit Forms: 1. 1

lim 1 1 x2 0

x

x

lim 1

ln a x

x

a,

for

a

0

x

lim 1 1 x2

x

x

lim 1 sin x x DNE

x

x

As you can see, this limit form can result in all limits from 0 to , and even DNE.

2. 0 0

lim x x0 x2

lim ax a , for any number a x0 x

lim x x0 x2

lim x DNE , lim x sin 1x DNE

x0 x2

x0 x

As you can see, this limit form can result in all limits from to , and even DNE.

3.

lim x2 x x

lim ax a , for a 0 x x

lim x 0 x x2

x2 lim x x

lim 2x xsinx DNE

x

x

As you can see, this limit form can result in all limits from to , and even DNE.

4. 0

lim x2 1

x

x

lim

x

a x

a

,

for

any

number

a

x

lim x2 1

x

x

lim

2x

xsinx

1

DNE

x

x

As you can see, this limit form can result in all limits from to , and even DNE.

5. 0

lim ex ln1x 0

x

lim

1

x

1 x

a,

for

0

a

1

x a

1

lim x x 1

x

1

lim ax x a , for a 1

x

1

lim xx x

x

1

lim

3

sin

x

x

x

DNE

x

As you can see, this limit form can result in all limits from 0 to , and even DNE.

6. 00

lim x 0 x0 3 l1n x

ln a

lim xlnx a , for a 0

x0

lim

1

1 x2

x

x0 2

1 x

xlim0 xx DNE , xlim0 x sin2 1x x DNE

As you can see, this limit form can result in all limits from 0 to , and even DNE.

7.

lim x x2

x

lim x a x a , for any number a

x

lim x2 x

x

lim x sin x x DNE

x

As you can see, this limit form can result in all limits from to , and even DNE.

L’Hopital’s Rule Guidelines:

Type of indeterminate form

Apply L’Hopital’s Rule to

1. lim f x 0 or lim f x

gx 0

g x

or lim f x whatever g x

lim f x gx

2. lim f x g x 0

lim f x or lim g x

1

1

gx

f x

3. lim f xgx =1 ,lim f xgx =00,lim f xgx =0

lim ln f x or lim g x

1

1

gx

ln f x

But remember to exponentiate to get the original limit.

4. lim f x g x 5. lim f x 0

g x whatever

lim g1x f 1x

1

gx f x

1

lim gx 1 f x

Warning: If the zeros of g x accumulate at a, then it might be the case that lim f x xa g x

appears to exist, but lim f x lim f x . xa g x xa g x

Example:

lim x cos xsin x has the form , and the limit doesn’t exist.

x esin x x cos xsin x

However,

lim x cos xsin x lim

1 sin2 x cos2 x

x esin x x cos xsin x

x esin x 1 sin2 x cos2 x cos xesin x x cos xsin x

.

lim

2cos2 x

x 2esin x cos2 x cos xesin x x cos xsin x

Since lim

2cos2 x

0 , the discontinuities in the function

x2 n1

2

2esin x

cos2

x

cos

xesin x

x

cos

x sin

x

2cos2 x

can be removed to yield the continuous function

2esin x cos2 x cos xesin x x cos xsin x

2cos x

;cos x 0

2cos x

2esin x cos x esin x x cos xsin x

, and lim

0.

x 2esin x cos x esin x x cos xsin x

0

;cos x 0

So a careless cancelation of cos x between the numerator and denominator would lead you to

believe that lim x cos xsin x 0 , when it actually doesn’t exist.

x esin x x cos xsin x

Some limits can be determined by inspection just by looking at the form of the limit – these predictable limit forms are called determinate. Other limits can’t be determined just by looking at the form of the limit and can only be determined after additional work is done – these unpredictable limit forms are called indeterminate.

Determinate Limit Forms:

Assuming that the functions involved in the limit are defined:

1. The limit forms a , for any number a, and bounded result in a limit of 0.

2. The limit form results in a limit of .

3. The limit form results in a limit of 0.

4. The limit form a , for 1 a 1, results in a limit of 0.

5. The limit form a , for a 1, results in a limit of .

6. The limit form a , for a 0 , results in a limit of .

7. The limit form a , for a 0 , results in a limit of 0.

8. The limit forms , , and a , for a 0 , result in a limit of . 0 a

9. The limit forms , , and a , for a 0 , result in a limit of . 0 a

10. The limit forms and result in a limit of .

11. The limit forms , , and result in a limit of .

12. The limit forms a 0 , for any number a, and bounded 0 result in a limit of 0.

13. The limit forms a , for any number a, and bounded result in a limit of .

14. The limit form a0 , for a 0, results in a limit of 1.

Indeterminate Limit Forms: 1. 1

lim 1 1 x2 0

x

x

lim 1

ln a x

x

a,

for

a

0

x

lim 1 1 x2

x

x

lim 1 sin x x DNE

x

x

As you can see, this limit form can result in all limits from 0 to , and even DNE.

2. 0 0

lim x x0 x2

lim ax a , for any number a x0 x

lim x x0 x2

lim x DNE , lim x sin 1x DNE

x0 x2

x0 x

As you can see, this limit form can result in all limits from to , and even DNE.

3.

lim x2 x x

lim ax a , for a 0 x x

lim x 0 x x2

x2 lim x x

lim 2x xsinx DNE

x

x

As you can see, this limit form can result in all limits from to , and even DNE.

4. 0

lim x2 1

x

x

lim

x

a x

a

,

for

any

number

a

x

lim x2 1

x

x

lim

2x

xsinx

1

DNE

x

x

As you can see, this limit form can result in all limits from to , and even DNE.

5. 0

lim ex ln1x 0

x

lim

1

x

1 x

a,

for

0

a

1

x a

1

lim x x 1

x

1

lim ax x a , for a 1

x

1

lim xx x

x

1

lim

3

sin

x

x

x

DNE

x

As you can see, this limit form can result in all limits from 0 to , and even DNE.

6. 00

lim x 0 x0 3 l1n x

ln a

lim xlnx a , for a 0

x0

lim

1

1 x2

x

x0 2

1 x

xlim0 xx DNE , xlim0 x sin2 1x x DNE

As you can see, this limit form can result in all limits from 0 to , and even DNE.

7.

lim x x2

x

lim x a x a , for any number a

x

lim x2 x

x

lim x sin x x DNE

x

As you can see, this limit form can result in all limits from to , and even DNE.

L’Hopital’s Rule Guidelines:

Type of indeterminate form

Apply L’Hopital’s Rule to

1. lim f x 0 or lim f x

gx 0

g x

or lim f x whatever g x

lim f x gx

2. lim f x g x 0

lim f x or lim g x

1

1

gx

f x

3. lim f xgx =1 ,lim f xgx =00,lim f xgx =0

lim ln f x or lim g x

1

1

gx

ln f x

But remember to exponentiate to get the original limit.

4. lim f x g x 5. lim f x 0

g x whatever

lim g1x f 1x

1

gx f x

1

lim gx 1 f x

Warning: If the zeros of g x accumulate at a, then it might be the case that lim f x xa g x

appears to exist, but lim f x lim f x . xa g x xa g x

Example:

lim x cos xsin x has the form , and the limit doesn’t exist.

x esin x x cos xsin x

However,

lim x cos xsin x lim

1 sin2 x cos2 x

x esin x x cos xsin x

x esin x 1 sin2 x cos2 x cos xesin x x cos xsin x

.

lim

2cos2 x

x 2esin x cos2 x cos xesin x x cos xsin x

Since lim

2cos2 x

0 , the discontinuities in the function

x2 n1

2

2esin x

cos2

x

cos

xesin x

x

cos

x sin

x

2cos2 x

can be removed to yield the continuous function

2esin x cos2 x cos xesin x x cos xsin x

2cos x

;cos x 0

2cos x

2esin x cos x esin x x cos xsin x

, and lim

0.

x 2esin x cos x esin x x cos xsin x

0

;cos x 0

So a careless cancelation of cos x between the numerator and denominator would lead you to

believe that lim x cos xsin x 0 , when it actually doesn’t exist.

x esin x x cos xsin x