Honors Calculus 9/27/16
Ch 2 REVIEW: Limits
Name
1. Evaluate each limit. Show all work.
a.
c.
lim
x →0
x+ 4 −2
x
1
1
+
lim x + 2 2
x →0
x
x2
e. (i) lim 2
x →6− x − 36
x2
(ii) lim 2
x →6+ x − 36
x2
(iii) lim 2
x →6 x − 36
x 3 + 27
x→−3 x + 3
b. lim
d.
x
x→0 tan x
lim
sin 6x
x→ 0
x
f. lim
⎧2 − 2x − x 2 , x ≤ 1
g. lim f (x) where f (x) = ⎨
x→1
,x >1
⎩ 2x − 3
Justify your answer.
⎛ x 2 − 2x −15 ⎞
i. lim ⎜ 2
⎟
x → 1+ ⎝ x − 6x + 5 ⎠
⎛ x 2 − 2x − 15 ⎞
h. lim ⎜ 2
x→5 ⎝ x − 6x + 5 ⎟
⎠
2. Given ƒ(x) =
x−3
, evaluate each limit. No work is necessary.
x−3
a. lim f ( x ) =
x → 3–
b. lim f ( x ) =
x → 3+
c. lim f ( x ) =
x→ 3
3. Let ƒ be defined as follows:
⎧⎪ x 2 − 5 , x ≤ −2
f ( x) = ⎨
⎪⎩ x + 2 , x > −2
Is ƒ(x) continuous? If not, describe the type of discontinuity (removable or nonremovable).
4. Multiple Choice: Let ƒ be defined as follows:
⎧ x 2 −1
⎪
f (x) = ⎨ x −1 for x ≠ 1
⎪⎩ 4
for x = 1
Which of the following statements must be true?
I. lim f (x) exists
x →1
II. ƒ(1) exists
III. ƒ(x) is continuous at x = 1
(A) I only
(B) II only (C) I and II only
(D) I and III only
(E) I, II, and III
5. Given the graph of ƒ(x),
find each limit.
a. lim f ( x ) =
x → −4
b. lim f ( x ) =
x→ 0
c. lim f ( x ) =
x→ 2
d. lim f ( x ) =
x→ 4
e. lim f ( x ) =
x → 4−
ƒ. Explain why ƒ(x) is not continuous at x = –4. Your justification must demonstrate your
knowledge of the definition of continuity at a point.
g. Describe the type of discontinuity (removable or nonremovable) at x = –4.
6. The functions ƒ and g are continuous functions.
The table above gives values of the functions at
selected values of x. The function h is given by
h(x) = ƒ(g(x)) – 6.
Explain why there must be a value r for 1 < r < 3
such that h(r) = –5.
x
ƒ(x)
g(x)
1
6
2
2
9
3
3
10
4
4
–1
6
7. The height, h (in feet), above the ground of a model rocket at time, t (in seconds), is given
by h(t) = −32t 2 + 160t + 192 . Use the intermediate value theorem to justify that the rocket
hits the ground between 5 and 7 seconds.
8. Sketch a function ƒ(x) such that ƒ(x) has nonremovable discontinuity at x = 2, lim f (x) = 4 , and
x→−3
removable discontinuity at x = –3.
9. Find the values of a and b that make the function ƒ below continuous. Show all work;
your answer should demonstrate your knowledge of the definition of continuity.
⎧ x+2 ,
⎪
ƒ( x) = ⎨ax 2 + bx + 3 ,
⎪ 2x − a + b ,
⎩
x<2
2≤x<3
3≤ x