Calculus H
Review 1.3-1.4
Name:_____________________
Date:__________Period:______
In Exercises 1-6, find all values x = a where the function is discontinuous. Identify each type of discontinuity.
Identify the condition of continuity that is not satisfied for each point of discontinuity.
1.
2.
3.
4.
5.
6.
Find all values x = a where the function is discontinuous. For each point of discontinuity, give
a) lim− f ( x)
b) lim+ f ( x)
c) lim f ( x)
d) f (a ) if it exists
7. f ( x) =
x →a
x →a
x →a
5+ x
x( x − 2)
10. f ( x) =
x+2
x+2
8. f ( x) =
9. f ( x) = x 2 − 4 x + 11
x2 − 4
x−2
11. f ( x) = e
x −1
In Exercises 13-14, a) graph the given function, b) find all values of x where the function is discontinuous,
and c)find the limit from the left and from the right at any values of x found in part b.
 1, x  2

12. f ( x) =  x + 3, 2  x  4
 7, x  4

x  −1
 11,
 2
13. g ( x) =  x + 2, − 1  x  3
 11,
x3

For 15-16, determine the indicated limits.
14.
a)
d)
−5,
x  −3

f ( x ) =  x, −3  x  3
 5,
x3

lim f ( x ) = _______
b)
lim− f ( x ) = _______
e)
x →−3−
15.
a)
d)
c)
lim+ f ( x ) = _______
f)
lim f ( x ) = _______
lim f ( x ) = _______
c)
lim f ( x ) = _______
lim f ( x ) = _______
f)
lim f ( x ) = _______
x →3
x →3
lim f ( x ) = _______
lim f ( x ) = _______
x →−3+
x →−3
x →3
 x,
x 1
 2
f ( x) = x , 1  x  2
 5,
x2

lim f ( x ) = _______
b)
lim f ( x ) = _______
e)
x →1−
x → 2−
x →1+
x → 2+
x →1
x →2
16. Use the Intermediate Value Theorem to justify that the function f ( x) = x 4 − 2 x 2 + 3x has a zero in the
interval  −2, −1 .
17. Use the Intermediate Value Theorem to justify that the function g ( x) = sin( x) has a value g (c) =
  3 
interval  ,  .
2 2 
3
in the
4
19-20, Find the constants a, or the constants a and b, such that the function is continuous on the entire real
line.
 x3 , x  2
19. f ( x) =  2
ax , x  2
x  −1
2,

20. f ( x) = ax + b, − 1< x  3
−2,
x3

21-34, Evaluate each limit.
x2 − 4
x →−2 x 2 + 8
22. lim
1 − cos(2 x)
x →0
4x
24. lim
21. lim
23. lim
25.
x →0
5 x 3 + 27
x →−2 20 x 2 + 10 x + 9
26. lim
x →1
27. lim
x→4
x +5 −3
x−4
sin11x
sin17 x
x −1
x −1
5cos 4 x − 5
x →0
x
28. lim
x+4
2
x →−4 x − 5 x − 36
2 x2 − x − 3
x →−1
x +1
29. lim
30. lim
sin 2 x
31. lim
x →0 tan 2 x
32. lim
33. lim+
x→
3
2
3 − 2x
15 − 10 x
x →0
34. lim−
x→
3
2
x + 16 − 4
x
3 − 2x
15 − 10 x