Calculus 1 Worksheet #5
Limits involving approaching infinity: lim f ( x)
x 
TO INFINITY AND BEYOND !!!!!
1
lim
Important theorem: x  x  0
Limits Involving Infinity
(Principle of Dominance)
xa
, if a  b. Then, limit = 0. (Look for the highest degrees/powers of x)
x  x b
C
Cx a
2. lim b , if a  b. Then, limit =
. (Look for the highest degrees/powers of x)
x  Dx
D
xa
3. lim b , if a  b. Then, limit =  or   . (Look for the highest degrees/powers of x
x  x
and check the sign of  by substituting with a large x–value.)
1. lim
Problems:
1. lim 7 
x 
1
2
 2
3x x
5x 2  2
x  4 x 2  7
3x3  2
9. lim
x  5 x 2  1
3  5x
13. lim
x  3 x  1
6. lim
17. lim
x
x 1
21. lim
5x2
x3
x 
x 
3x3  5
x  5 x 2  1
3x 2  2
10. lim
x  4 x 2  1
3  2 x2
14. lim
x 
3x  1
5. lim
x 
4x  8
5x
2. lim
2
8 x 2  3x
x 
2 x2  1
1
4
22. lim x  2
x  2
x
18. lim
3. lim
x 
3x  1000
x  100
2 x2  4 x
x 
x 1
x2  2
11. lim
x  x  555
15.
6 x2  2 x 1
lim
x  2 x 2  3 x  2
2
19. lim 10  2
x 
x
sin x
23. lim
x 
x
7. lim
4. lim
x 
5x  5
7 x2  1
2x2  4x
x 
x 1
3  2x
12. lim
x  3 x 3  1
8. lim
16. lim
x 
3x3  2
2 x 2  9 x3  7
3
x 
x
cos 2 x
24. lim
x 
3x
20. lim 4 
Answers:
1) 7
7) 
5
3
19) 10
13) –
4
5
8) – 
3) 3
14) 
15) 3
20) 4
21) 
2)
9) 
4) 0
3
4
1
16)
3

22)
10)
5
4
11) 
6) 
17) 0
18) 4
23. 0
24. 0
5)
12) 0
Revised: 7/13/2017