Math 151 Week in Review
Monday Sept. 20, 2010
Instructor: Jenn Whitfield
Thanks to Amy Austin for contributing some problems.
All prolbems in this set are copywrited
Section 2.5
• Definition Continuous: A function f (x) is
continuous at a number a if lim f (x) = f (a).
x→a
Section 2.3
• If f is a polynomial or a rational function and a is in the domain of f , then
lim f (x) = f (a)
x→a
• The
Squeeze
Theorem - If
f (x) ≤ g(x) ≤ h(x) for all x in an open
interval that contains a (except possibly
at a) and lim f (x) = lim h(x) = L then
x→a
x→a
lim g(x) = L
x→a
1. Given lim f (x) = 7, lim g(x) = −1 and
x→2
x→2p
• Definition Continuous from the Right: A
function f (x) is continuous from the right at a
number a if lim+ f (x) = f (a) .
x→a
• Definition Continuous from the Left: A
function f (x) is continuous from the left at a
number a if lim f (x) = f (a) .
x→a−
• The Intermediate Value Theorem: Suppose
that f is continuous on the closed interval [a, b]
and let N be any number strictly between f (a)
and f (b). Then there exits a number c in (a, b)
such that f (c) = N .
1
8 g(x)h(x)
.
lim h(x) = − , find lim
x→2
x→2 (f (x) + g(x))2
2
x2 − x − 2
2. Find lim 2
.
x→−1 x + 3x + 2
√
x+1−4
.
3. Find lim
x→15
x − 15
4. Find lim
x→3
1 1
−
x 3
5. Find lim ~r(t), where
1
x−3
11. Referring to the graph below, determine the
values of x where f (x) is discontinuous and
then state the mathematical reason why the
function is discontinous.
f(x)
6
.
t→−2
t2
−4
−2
2
4
6
8
x
−2
−4
−6


 x
x<0
12. Let f (x) =
0 ≤ x ≤ 1 and show

 1 x>1
x3
that f is continuous on (−∞, ∞).
(a) lim f (x)
x2
x→3+
(b) lim f (x)
x→3−
(c) lim f (x)
x→3
| 3x − 6 |
.
7. Find lim
x→2 x − 2
3
1
9. Given − x + 2 ≤ f (x) ≤ for 0 < x ≤ 6,
3
x
what is lim f (x)?
x→3
10. Find all values of a such that the following
limit exists:
2x2 − 3ax + x − a − 1
lim
x→1
x2 + 2x − 3

3

 4 − 5x
x<5
4
x=5

 −1 + x x > 5
answer the questions below.
13. Given f (x) =
10
8. Find lim 2x cos
.
x→0
x
2
2
−6
t2
−4 ~
j
− 5t − 14
( √
x2 + 16 if x ≤ 3
6. Given f (x) =
, find
3
x − 10
if x > 3
the following:
~r(t) = 4t2~i +
4
(a) At x = 5, is f (x) continuous from the
right?
(b) At x = 5, is f (x) continuous from the
left?
(c) Find lim f (x)
x→5
(d) Is f (x) continuous at x = 5?
14. Find the values of c and d that will make
f (x) =


 dx − c
if x ≤ 0
cx + d
if 0 < x ≤ 3

 x2 − dx − 11 if x > 3
continuous on all real numbers.
15. Use the Intermediate Value Theorem
√ to
2
show that there is a root of x = x + 1
on (1, 2).
Section 2.6
• If r > 0 is a rational number such that xr is
1
defined for all x, then lim r = 0.
x→±∞ x
•
√
x2 = |x| =
(
x
if x > 0
.
−x if x ≤ 0
• Horizontal Asymptotes
lim f (x).
are
lim f (x)
x→∞
x→−∞
16. Compute the following limits:
2x2 + 5
.
x→∞ 3x5 + 1
4x3 − 6x4
(b) lim
.
x→∞ 2x4 − 9x + 1
t9 − 4t10
.
(c) lim 8
t→∞ t + 2t2 + 1
(d) lim ~r(t) where
(a) lim
t→∞
~r(t) =
*
√
t2 − 9t + 3
,
t − t3 + 1
2 + 25x2
.
x→∞
4 − 3x
√
3x2 + 1
.
(f) lim
x→−∞ 4x − 3
(e) lim
17. Find all phorizontal
f (x) = x − x2 + x.
√
+
t+1
.
t
asymptotes
for
and