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:
a bounded 1. The limit forms , for any number a, and 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 11 a , results in a limit of 0.
5. The limit form , 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 , result in a limit of . 0 a
9. The limit forms , , and a, for , result in a limit of . 0 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 boundedresult in a limit of .
14. The limit form a0 , for a 0, results in a limit of 1. Indeterminate Limit Forms:
1. 1
2 1 x lim 1x 0 x
ln a x lim 1x a , for a 0 x
2 1 x lim 1x x
sin x x lim 1x DNE x
As you can see, this limit form can result in all limits from 0 to , and even DNE.
0 2. 0
x lim 2 x0 x
ax lim a , for any number a x0 x
x lim 2 x0 x
x xsin 1 lim DNE , lim x DNE x0 x2 x0 x
As you can see, this limit form can result in all limits from to , and even DNE.
3.
x2 lim x x
ax lim a , for a 0 x x
x lim 0 x x2
x2 lim x x
2x xsinx lim DNE x x
As you can see, this limit form can result in all limits from to , and even DNE.
4. 0
2 1 lim x x x
a lim xax , for any number a x
2 1 lim x x x
1 lim 2x xsinx DNE x x
As you can see, this limit form can result in all limits from to , and even DNE.
5. 0
1 limex ln x 0 x
1 1 x x lim a a , for 01a x
1 limx x 1 x
1 lim aax x , for a 1 x
1 lim xx x x
1 lim 3 sin xx x DNE x
As you can see, this limit form can result in all limits from 0 to , and even DNE.
6. 00
1 limx3 ln x 0 x0
ln a lim xaln x , for a 0 x0
1 x 1 x2 lim 2 x0
1 x x 2 1 x lim x DNE , limx sin x DNE x0 x0
As you can see, this limit form can result in all limits from 0 to , and even DNE.
7.
lim xx2 x
lim x a x a , for any number a x
lim xx2 x
limx 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
fx 0 fx 1. lim or lim gx 0 gx fx lim gx fx whatever or lim gx
f x g x 2. limf x g x 0 lim or lim 11 g x f x
ln f x g x lim or lim 11 g x g x 00 g x g x ln f x 3. limf x =1 ,lim f x =0 ,lim f x = But remember to exponentiate to get the original limit.
11 4. limf x g x limg x f x 1 g x f x
1 fx 0 gx 5. lim lim 1 g x whatever fx
fx Warning: If the zeros of gx accumulate at a, then it might be the case that lim xa gx f x f x appears to exist, but lim lim . x ag x x a g x
Example:
x cos x sin x lim has the form , and the limit doesn’t exist. However, x esin x x cos x sin x
x cos x sin x 1 sin22xx cos lim lim sinxx 2 2 sin xxsin x e1 sin x cos x cos xe x cos x sin x e x cos x sin x . 2cos2 x lim x 2esinxx cos 2 x cos xe sin x cos x sin x 2cos2 x Since lim 0 , the discontinuities in the function 21n sinxx 2 sin x 2 2e cos x cos xe x cos x sin x 2cos2 x can be removed to yield the continuous function 2esinxx cos 2 x cos xe sin x cos x sin x 2cos x sinxx sin ;cosx 0 2cos x 2e cos x e x cos x sin x , and limsinxx sin 0. x 2e cos x e x cos x sin x 0 ;cosx 0 So a careless cancelation of cos x between the numerator and denominator would lead you to x cos x sin x believe that lim 0, when it actually doesn’t exist. x esin x x cos x sin x