Math 147 Review
Limits and Continuity
Definition: If the values of a function f (x) can be made arbitrarily close to a real number L
by taking x sufficiently close (but not equal to) a, then we say that the limit of f (x), as x
approaches a, equals L. This is denoted by
lim f (x) = L.
x→a
Similar definitions hold for one-sided limits and limits at infinity.
Example: Evaluate each limit.
(a) lim (3x4 − 2x + 1)
x→2
(b) lim 3 cos
x→π
x
4
x2 − 2x − 3
x→3
x−3
(c) lim
√
(d) lim
x→0
x2 + 9 − 3
x
1
(e) lim+
1
x−3
(f) lim+
4
2−x
x→3
x→2
2x2 − 3x + 5
(g) lim 4
x→∞ x − 2x + 1
6x2 + 2x + 1
x→−∞ 3x2 + x − 4
(h) lim
x2 − 2
x→∞ x + 1
(i) lim
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Definition: A function f is continuous at x = a if
lim f (x) = f (a).
x→a
In order for f to be continuous at x = a, the following conditions must hold:
1. f (a) is defined
2. lim f (x) exists
x→a
3. lim f (x) = f (a)
x→a
If f is not continuous at x = a, then f is said to be discontinuous at x = a.
Example: Consider the function defined by

 x2 + x − 2
if x 6= 1
f (x) =
x
−
1

a
if x = 1
Find the value of a that makes f continuous on R = (−∞, ∞).
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Differentiation
Definition: The derivative of f (x) is given by
f (x + h) − f (x)
h→0
h
f 0 (x) = lim
provided that this limit exists. If the limit exists, then f is called differentiable.
Example: Differentiate each function using the rules of differentiation.
(a) f (x) = 4x3 − 7x + 1
N
(b) f (N ) = rN 1 −
K
(c) f (x) = (3x4 − 5)(2x − 5x3 )
(d) f (x) =
3x2 − 2x
2x + 1
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(e) f (x) =
√
3x2 + 5x
(f) f (x) = 2 sin(3x2 + 5) − 3 cos(1 − 2x) + 5 tan(4x)
(g) f (x) = e4x
2 −2x+1
(h) f (x) = ln(2x3 − x)
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Example: Find an equation of the tangent line to the curve defined implicitly by
xy − y 3 = 1
at the point (2, 1).
Example: A ladder 10 feet long is leaning against a vertical wall. The base of the ladder is
pulled away from the wall at a rate of 3 feet/second. How fast is the top of the ladder sliding
down the wall when the base is 6 feet from the wall?
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Antidifferentiation/Integration
Definition: A function F is called an antiderivative of f on an interval if F 0 (x) = f (x) for
all x in that interval. If F is an antiderivative of f on an interval, then the most general
antiderivative is denoted by
Z
f (x) dx = F (x) + C.
This is called an indefinite integral.
Example: Evaluate each indefinite integral.
Z
(a)
(x3 − 3x2 + 2x − 4) dx
Z 1
x
(b)
sin x + + e
dx
x
Z
(c)
(cos 3x + ex/2 ) dx
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Definition: Consider a continuous function f defined on a closed, bounded interval [a, b]. Let
P = {x0 , x1 , x2 , . . . , xn } be a partition of [a, b] where
a = x0 < x1 < x2 < · · · < xn = b.
Choose a representative point x∗k in each subinterval [xk−1 , xk ] and let ∆xk = xk − xk−1 and
||P || = max{∆xk }. Then the definite integral of f from a to b is given by
Z
b
f (x) dx = lim
a
||P ||→0
n
X
f (x∗k )∆xk ,
k=1
provided that this limit existst. If the limit does exist, then f is called integrable on [a, b].
Note: By the Fundamental Theorem of Calculus,
Z b
f (x) dx = F (x)|ba = F (b) − F (a),
a
where F is any antiderivative of f .
Example: Evaluate each definite integral.
Z 4
(a)
(3 − 2x) dx
2
Z
(b)
1
(x3 − x1/3 ) dx
0
Z
(c)
π/4
sin(2x) dx
0
8
Z
1
(d)
x2
0
Z
1
dx
+1
1
|x| dx
(e)
−1
Example: Find the area under the graph of y = ex from x = 0 to x = 2.
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