Math 100 - Homework Set 1 (Limits and Continuity)
due date: Thursday February the 4th at 8am (in class or by e-mail)
Basic Skills required to work through the problems:
• factoring quadratic and cubic polynomials;
• manipulating (factoring, simplifying, razionalizing, etc.) expressions with ratios
and square roots of polynomials;
• interpreting and manipulating expressions containing absolute values;
• factoring and simplifying expressions containing ratios and square roots of polynomials;
• determining the domain of a function;
Learning Goals: After completing this set, you should be able to:
• master the basic skills listed above;
• compute the limit of a function at a point using limit laws and algebraic manipulation techniques, or determine that such limit does not exists.
• compute the limit of a function at a point using limit laws, algebraic manipulation
techniques, or the squeeze theorem, or determine that such limit does not exist;
• evaluate the limit of a function at infinity using limit laws and algebraic manipulation techniques, or determine that such limit does not exist;
• determine if a function has horizontal or vertical asymptotes, and find their equation;
• identify whether a function is discontinuous and, if possible, find conditions for
which the discontinuity can be eliminated;
• apply the Intermediate Value Theorem, use it to construct simple proofs about a
given statement.
1. A warm up. Write out factorizations of the following expressions:
(a) x2 − 25
(b) x2 − x − 2
(c) x3 − 8
2. More warm up. Simplify:
x2 − 25
(a)
x+5
x2 − x − 2
(b)
x−2
(c)
x2 − 4
x3 − 8
3. Evaluate the following limits.
x2 − 25
(a) lim
x→−5 x + 5
x2 − 25
(b) lim
x→−1 x + 5
x2 − x − 2
(c) lim
x→2
x−2
2
x −4
(d) lim 3
x→2 x − 8
4. Evaluate the following limit:
1
+ 12
x
x→−2 x3 + 8
lim
5. Evaluate the following limit:
lim
x→0
|2x − 2| − |2x + 2|
x
6. (a) What is the domain of the function g(t) =
√1
t 1+t
−
1
t
?
(b) Evaluate the following limit:
lim
t→0
7. Evaluate the following limit:
1
1
√
−
t 1+t t
√
25 + h − 4
lim
h→0
h
8. True or False. If false, give a counter-example.
(a) If f (−1) = −1 and f (1) = 1, then f (0) = 0.
(b) If f (−1) = −1 and f (1) = 1, then there is a point c such that −1 < c < 1 and
f (c) = 0.
(c) If f (−1) = −1 and f (1) = 1, and f (x) is continuous then there is a point c
such that −1 < c < 1 and f (c) = 0.
9. Use the Intermediate Value Theorem to show that the function f (x) = x3 + 2x2 −
4x − 1 has at least three zeros in the interval [−4, 4].
10. Consider the following function, where a is an unspecified parameter:
(
f (x) =
(x − 3)2 + a, x ≥ 1;
1
+ 2a − 3, x < 1.
2x
Then f (x) could be discontinuous for, at most, two values of x. Identify these
x-values. Choose a so that one of these discontinuities disappears.
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4 − 3x
.
x+7
(a) Find the following limits: (i) lim f (x), (ii) lim f (x), (iii) lim f (x).
11. Let f (x) =
x→∞
x→−∞
x→−7
(b) Specify any horizontal or vertical asymptotes (if any).
(c) Roughly sketch the graph of f (x) using the information you found in parts (a)
and (b).
12. (Optional) Show by means of example that lim [f (x) + g(x)] may exist even though
x→a
neither lim f (x) nor lim g(x) exists.
x→a
x→a
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