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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
3×3?1 x = (c) limx?0
7×2?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??
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 ??0
?2 cos (
1 ?
) (b) lim
x?0 ex?1
x
5. Consider the function: f (x) = ?2×3 ? 7×2 + 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 ???
sin ? tan ? (b) lim
h?0
1 a+h?
1 a
h (c) lim
???2
tan ??tan a 1+tan ? 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.
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