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ProblemSet2.pdf
Name: The Limit of a Function Section:
2.2 The Limit of a Function 1. Determine the following limits.
2 4
2
4
x
f (x)
−5 5 −2
2
4
x
f (x)
1 2 3 −2
2
4
x
f (x)
lim x→4
f (x) = lim x→−3
f (x) = lim x→∞
f (x) =
lim x→3
f (x) = lim x→0
f (x) = lim x→0
f (x) =
lim x→2
f (x) = lim x→2
f (x) =
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Name: The Limit of a Function Section:
2. Determine the following limits.
(a) lim x→−8
− 1 3 x
5 − x1/3 = (b) lim x→ 3 √
1/3
3x3−1 x = (c) limx→0
7x2−35 x−5 =
(d) lim x→−11
(x+1)(x+11) x+11 = (e) lim
x→2 x2+4x−12
x−2 = (f) limx→0+ ln xx+2 ln x2
=
3. Sketch and carefully label a graph that has all of the following limits.
lim x→1
f (x) = 2 lim x→−4
f (x) = 2 lim x→0
f (x) = 12 limx→5 f (x) does not
exist
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Name: The Limit of a Function Section:
4. Use the graph to determine the following:
(a) lim x→1
f (x) =
(b) lim x→2+
f (x) =
(c) lim x→0
f (x) =
(d) lim x→1+
f (x) =
(e) lim x→1−
f (x) =
(f) lim x→−1
f (x) =
−1 1 2 3 4
−2
2
x
f (x)
5. Determine the following limits.
(a) lim x→0
xe2x−xex ex−1
(b) lim x→π
sin(2x) sin x (c) limx→0
sin (
1 x
)
6. Determine the following limits.
(a) lim x→7
3 x−7 (b) limx→2
5 (2−x)2
(c) lim x→0
x sin x
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Name: The Limit Laws Section:
2.3 The Limit Laws Limit Laws
Limit of the Identity Function
lim x→a
x =
Limit of the Constant Function
For a constant c, lim x→a
c =
Sum Law of Limits lim
x→a ( f (x) + g(x)) =
Difference Law of Limits lim
x→a ( f (x) − g(x)) =
Constant Multiple Law for Limits
lim x→a
(c · f (x)) =
Product Law for Limits lim
x→a ( f (x) · g(x)) =
Quotient Law for Limits lim
x→a
( f (x) g(x)
) =
Power Law for Limits lim
x→a ( f (x))n =
Squeeze Theorem Given functions f , g, and h such that ≤ ≤ ,
if = = L, then
lim x→a
g(x) =
Special Limits
lim x→0
ex−1 x = limx→0
sin x x = limx→0
cos x−1 x =
1. Find the following limits.
(a) lim x→−1
x2+5x x4+2
(b) lim x→1
x−1 x2−1 (c) lim
h→0 (3+h)2−9
h
(d) lim t→0
√ t2+9−3
t2 (e) limx→0 |x|
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Name: The Limit Laws Section:
2. Find the following limits:
(a) lim h→0
(7+h)2−49 h (b) limh→0
(−3+h)2−(−3)2 h (c) limh→0
(4+h)2+2−(42+2) h
(d) lim h→0
(8+h)7−(8)7 h (e) lim
h→0 (−1+h)3−(−1+h)2+17−((−13−(−1)2+17))
h
3. Find the following limits:
(a) lim x→3
x2−6x+9 √
x−3 (b) lim
c→1 c2−c √
c−1 (c) lim
p→1
1−p √
3−p− √
2
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Name: The Limit Laws Section:
4. Use the squeeze theorem to determine the following limits.
(a) lim θ→0
θ2 cos (
1 θ
) (b) lim
x→0 ex−1
x
5. Consider the function: f (x) = −2x3 − 7x2 + 1. Determine the following limits.
(a) lim h→0
f (h)− f (0) h (b) limh→0
f (x+h)− f (x) h
6. Find the following limits:
(a) lim θ→π
sin θ tan θ (b) lim
h→0
1 a+h−
1 a
h (c) lim
θ→π2
tan θ−tan a 1+tan θ tan a
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Name: Continuity Section:
2.4 Continuity Composite Functions Theorem
If f (x) is continuous at L and lim x→a
g(x) = L, then:
lim x→a
f (g(x)) = =
Intermediate Value Theorem
For any closed, bounded interval [a, b], if z is a real number between and
, then there exists a number c in [a, b] such that f (c) =
1. Evaluate lim x→0
ln (
sin x x
) 2. Evaluate lim
x→0 sin
( ex−1
x
)
3. Determine whether each of the following functions is continuous over its domain. If it is not, state where it is discontinuous.
1
1
2
x
f (x)
1
−2
2
4
x
f (x)
1 −2
2
4
x
f (x)
4. Determine whether each of the following functions is continuous over its domain. If it is not, state where it is discontinuous.
(a) f (x) = 1 x2−1
(b) f (x) = 4 x2+1 (c) f (x) =
|x−2| x−2
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Name: Continuity Section:
5. For each of the following, determine whether the function is continuous at the given point. If it is not, state what type of discontinuity it is.
(a) f (x) = 2x 2−5x+3 x−1 at x = 1
(b) f (y) = sin(πy)tan(πy) ta y = 1
(c) f (x) = {
x2 − ex ; x < 0 x − 1 ; x ≥ 0
at x = 0
(d) h(θ) = sin θ−cos θtan θ at θ = π
6. For each of the following functions and intervals, determine whether or not the Intermediate Value Theorem applies and whether or not given value of f can be found on the interval.
(a) f (x) = 2x|x−1|3x−3 , f (x) = 0 on [0, 2]
(b) f (x) = x 3+x2−6x
x−2 , f (x) = 2 on [−4, 0]
(c) f (x) = x 3+x2−6x
x−2 , f (x) = −2 on [−4, 0]
7. Use the Intermediate Value Theorem to determine whether the equation 2x = x3 has a solution over either of the intervals [1.25, 1.375] or [1.375, 1.5]. Explain your answer for each interval.
8. Suppose that y = f (x) is defined for all x. Sketch a graph that meets the conditions listed for each of the following.
(a) f is discontinuous at x = 1 with lim
x→−1 f (x) = −1 and
lim x→2
f (x) = 4
(b) f is only discontinuous at x = 2 and lim
x→0 = 12
(c) f is discontinous at x = 0, with lim
x→2+ f (x) = 2 and
lim x→2−
f (x) = −1
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Name: Limits Practice I Section:
A.2 Limits Practice I Evaluate the following limits.
1. lim x→2
3x5−12 x−120 x−1
2. lim x→0
log3(x + 243) 3. lim x→0
sin ( x + π6
)
4. lim x→0
7x5−2x3+x x 5. limx→−2
4x2−6x−28 x+2 6. limx→−5/3
3x3+5x2−6x−10 5 x +3
7. lim x→0
sin(x+π) x 8. lim
x→0 x2+6x+9 √
x+3 9. lim
h→0
√ 27+h−
√ 27
h
Sketch a graph that satisfies the given criteria.
10. lim x→−∞
f (x) = −2, lim x→∞
f (x) = 2, lim
x→−4+ f (x) = ∞, lim
x→2 f (x) = 5
11. lim x→−∞
f (x) = ∞, lim x→∞
f (x) = −∞, lim
x→−5+ f (x) = −10, lim
x→1 f (x) = 5
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