Limits 7 Worksheet
Calculus
Name___________________________________________
Find the limit (if it exists).
1. lim x®-1
x 2 +1
x
2. lim x®3
x +1
x-4
3. lim x®p tan x
4. lim x®-1
x 2 -1
x +1
5. lim x®-1
2x 2 - x - 3
x +1
6. lim Dx®0
7. lim Dx®0
(x + Dx)2 - x 2
Dx
8. lim x®0
10. lim x®0
sin 2 x
x
11. lim x®0
sin 2x
sin 3x
14. lim x®¥ sin
13. lim x®0
16. lim x®5+
x -5
x 2 - 25
19. lim x®p cot x
sin x
5x
9. limq ®0
tan 2 x
x
12. lim
1
x
cosq tan q
q
p
2
cos x
cot x
15. lim x®0 sin
2- x
x2 - 4
17. lim x®2+
x®
2(x + Dx) - 2x
Dx
18. lim x®0
1
x
x
x
21. lim x®2+
x-3
x-2
2x 2
5x 2 +16
24. lim x®¥
3x 2 - 6x - 7
x +1
(
27. lim x®0 x 3
(
)
20. lim x®3+ 2 éë x ùû -1
22. lim x®1+
2+ x
1- x
23. lim x®¥
25. lim x®¥
x-2
x2
26. lim x®2 5x - 3
)
__________________________________________________________________________________________
Find the constants a and b such that the function is continuous on the entire real line.
3
ïì x , x £ 2
28. f (x) = í 2
ïîax , x > 2
ì 4sin x
,x £ 0
ï
29. f (x) = í x
ïîa - 2x, x > 0
ì x2 - a2
ï
,x ¹ a
30. f (x) = í x - a
ï8, x = a
î
Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed
by the theorem.
31. f (x) = x 2 + x -1,[0,5], f (c) = 11
32. f (x) = x 2 - 6x + 8,[0,3], f (c) = 0
Determine the vertical and horizontal asymptotes.
33. f (x) =
3
x-2
34. f (x) =
x 2 +1
2x 2 -1
36. f (x) =
x 3 -1
x2
Determine the End Behavior Model.
35. f (x) =
x 3 -1
x -1
37. The rate at which ice is melting in a pond is given by R(t) = 1+ 2t , where R is the rate of change of ice
in cubic feet, and t is the time in minutes. Estimate the amount of ice that has melted in the first 5 minutes.
38. Suppose a car is moving with increasing speed according to the following table. Estimate the approximate
distance traveled in the first 10 seconds.
time(sec)
0
2
4
6
8
10
speed( ft / sec) 30 36 40 48 54 60