Applied Calculus I
Lecture 6
Defining limits of functions
Definition. A real number L is called (two-sided) limit of a function f
at a point a, denoted lim f (x) = L, if the values f (x) can be made
x→a
arbitrarily close to L by choosing x sufficiently close to a.
Defining limits of functions
Definition. A real number L is called (two-sided) limit of a function f
at a point a, denoted lim f (x) = L, if the values f (x) can be made
x→a
arbitrarily close to L by choosing x sufficiently close to a.
L
a
Defining limits of functions
Definition. A real number L is called (two-sided) limit of a function f
at a point a, denoted lim f (x) = L, if the values f (x) can be made
x→a
arbitrarily close to L by choosing x sufficiently close to a.
L
a
Defining limits of functions
Definition. A real number L is called (two-sided) limit of a function f
at a point a, denoted lim f (x) = L, if the values f (x) can be made
x→a
arbitrarily close to L by choosing x sufficiently close to a.
L
a
Defining limits of functions
Definition. A real number L is called (two-sided) limit of a function f
at a point a, denoted lim f (x) = L, if the values f (x) can be made
x→a
arbitrarily close to L by choosing x sufficiently close to a.
L
a
Defining limits of functions
Definition. A real number L is called (two-sided) limit of a function f
at a point a, denoted lim f (x) = L, if the values f (x) can be made
x→a
arbitrarily close to L by choosing x sufficiently close to a.
f (a)
L
L
a
a
Defining limits of functions
Definition. A real number L is called (two-sided) limit of a function f
at a point a, denoted lim f (x) = L, if the values f (x) can be made
x→a
arbitrarily close to L by choosing x sufficiently close to a.
f (a)
L
L
a
?
a
Defining limits of functions
Definition. A real number L is called (two-sided) limit of a function f
at a point a, denoted lim f (x) = L, if the values f (x) can be made
x→a
arbitrarily close to L by choosing x sufficiently close to a.
f (a)
L
L
?
a
a
Definition. A real number L is called the limit of a function f at a
point a from the left, denoted lim− f (x) = L, if the values f (x) can be
x→a
made arbitrarily close to L by choosing x sufficiently close to a and less
than a.
Definition. A real number L is called the limit of a function f at a
point a from the right, denoted lim+ f (x) = L, if the values f (x) can be
x→a
made arbitrarily close to L by choosing x sufficiently close to a and
greater than a.
Existence of limits
lim f (x) does not exist if:
x→a
Existence of limits
lim f (x) does not exist if:
x→a
• The values f (x) become arbitrarily large in magnitude (positive or
negative) as x approaches a. In the case of positive values we write
lim f (x) = ∞, In the case of negative values we write
x→a
lim f (x) = −∞
x→a
Existence of limits
lim f (x) does not exist if:
x→a
• The values f (x) become arbitrarily large in magnitude (positive or
negative) as x approaches a. In the case of positive values we write
lim f (x) = ∞, In the case of negative values we write
x→a
lim f (x) = −∞
x→a
• The values f (x) become arbitrarily large in magnitude as x
approaches a, but positive as x approaches a from one side, and
negative as x approaches a from the other side.
Existence of limits
lim f (x) does not exist if:
x→a
• The values f (x) become arbitrarily large in magnitude (positive or
negative) as x approaches a. In the case of positive values we write
lim f (x) = ∞, In the case of negative values we write
x→a
lim f (x) = −∞
x→a
• The values f (x) become arbitrarily large in magnitude as x
approaches a, but positive as x approaches a from one side, and
negative as x approaches a from the other side.
• One-sided limits exist but are not equal.
Existence of limits
lim f (x) does not exist if:
x→a
• The values f (x) become arbitrarily large in magnitude (positive or
negative) as x approaches a. In the case of positive values we write
lim f (x) = ∞, In the case of negative values we write
x→a
lim f (x) = −∞
x→a
• The values f (x) become arbitrarily large in magnitude as x
approaches a, but positive as x approaches a from one side, and
negative as x approaches a from the other side.
• One-sided limits exist but are not equal.

 |x|
, x 6= 0
f (x) =
x

0, x = 0
1
−1
Existence of limits
lim f (x) does not exist if:
x→a
• The values f (x) become arbitrarily large in magnitude (positive or
negative) as x approaches a. In the case of positive values we write
lim f (x) = ∞, In the case of negative values we write
x→a
lim f (x) = −∞
x→a
• The values f (x) become arbitrarily large in magnitude as x
approaches a, but positive as x approaches a from one side, and
negative as x approaches a from the other side.
• One-sided limits exist but are not equal.

 |x|
, x 6= 0
f (x) =
x

0, x = 0
lim− f (x) = −1,
x→0
1
−1
lim+ f (x) = 1
x→0
Rules for finding limits
Suppose that lim f (x) = A and lim g(x) = B.
x→a
x→a
Rules for finding limits
Suppose that lim f (x) = A and lim g(x) = B.
x→a
x→a
1. If k is a constant then lim kf (x) = k lim f (x) = kA
x→a
x→a
Rules for finding limits
Suppose that lim f (x) = A and lim g(x) = B.
x→a
x→a
1. If k is a constant then lim kf (x) = k lim f (x) = kA
x→a
x→a
2. lim (f (x) ± g(x)) = lim f (x) ± lim g(x) = A ± B
x→a
x→a
x→a
Rules for finding limits
Suppose that lim f (x) = A and lim g(x) = B.
x→a
x→a
1. If k is a constant then lim kf (x) = k lim f (x) = kA
x→a
x→a
2. lim (f (x) ± g(x)) = lim f (x) ± lim g(x) = A ± B
x→a
x→a
x→a
3. lim (f (x) · g(x)) = lim f (x) · lim g(x) = A · B
x→a
x→a
x→a
Rules for finding limits
Suppose that lim f (x) = A and lim g(x) = B.
x→a
x→a
1. If k is a constant then lim kf (x) = k lim f (x) = kA
x→a
x→a
2. lim (f (x) ± g(x)) = lim f (x) ± lim g(x) = A ± B
x→a
x→a
x→a
3. lim (f (x) · g(x)) = lim f (x) · lim g(x) = A · B
x→a
x→a
x→a
limx→a f (x)
A
f (x)
=
=
4. If B 6= 0, lim
x→a g(x)
limx→a g(x)
B
Rules for finding limits
Suppose that lim f (x) = A and lim g(x) = B.
x→a
x→a
1. If k is a constant then lim kf (x) = k lim f (x) = kA
x→a
x→a
2. lim (f (x) ± g(x)) = lim f (x) ± lim g(x) = A ± B
x→a
x→a
x→a
3. lim (f (x) · g(x)) = lim f (x) · lim g(x) = A · B
x→a
x→a
x→a
limx→a f (x)
A
f (x)
=
=
4. If B 6= 0, lim
x→a g(x)
limx→a g(x)
B
5. If p(x) is a polynomial then lim p(x) = p(a)
x→a
Rules for finding limits
Suppose that lim f (x) = A and lim g(x) = B.
x→a
x→a
1. If k is a constant then lim kf (x) = k lim f (x) = kA
x→a
x→a
2. lim (f (x) ± g(x)) = lim f (x) ± lim g(x) = A ± B
x→a
x→a
x→a
3. lim (f (x) · g(x)) = lim f (x) · lim g(x) = A · B
x→a
x→a
x→a
limx→a f (x)
A
f (x)
=
=
4. If B 6= 0, lim
x→a g(x)
limx→a g(x)
B
5. If p(x) is a polynomial then lim p(x) = p(a)
x→a
k
6. For any real number k, lim (f (x))k = lim f (x) , if the limit exists.
x→a
x→a
Rules for finding limits
Suppose that lim f (x) = A and lim g(x) = B.
x→a
x→a
1. If k is a constant then lim kf (x) = k lim f (x) = kA
x→a
x→a
2. lim (f (x) ± g(x)) = lim f (x) ± lim g(x) = A ± B
x→a
x→a
x→a
3. lim (f (x) · g(x)) = lim f (x) · lim g(x) = A · B
x→a
x→a
x→a
limx→a f (x)
A
f (x)
=
=
4. If B 6= 0, lim
x→a g(x)
limx→a g(x)
B
5. If p(x) is a polynomial then lim p(x) = p(a)
x→a
k
6. For any real number k, lim (f (x))k = lim f (x) , if the limit exists.
x→a
x→a
7. lim f (x) = lim g(x) if f (x) = g(x) for all x 6= a
x→a
x→a
Rules for finding limits
Suppose that lim f (x) = A and lim g(x) = B.
x→a
x→a
1. If k is a constant then lim kf (x) = k lim f (x) = kA
x→a
x→a
2. lim (f (x) ± g(x)) = lim f (x) ± lim g(x) = A ± B
x→a
x→a
x→a
3. lim (f (x) · g(x)) = lim f (x) · lim g(x) = A · B
x→a
x→a
x→a
limx→a f (x)
A
f (x)
=
=
4. If B 6= 0, lim
x→a g(x)
limx→a g(x)
B
5. If p(x) is a polynomial then lim p(x) = p(a)
x→a
k
6. For any real number k, lim (f (x))k = lim f (x) , if the limit exists.
x→a
x→a
7. lim f (x) = lim g(x) if f (x) = g(x) for all x 6= a
x→a
x→a
8. For any real number b > 0, lim b
x→a
f (x)
=b
limx→a f (x)
Rules for finding limits
Suppose that lim f (x) = A and lim g(x) = B.
x→a
x→a
1. If k is a constant then lim kf (x) = k lim f (x) = kA
x→a
x→a
2. lim (f (x) ± g(x)) = lim f (x) ± lim g(x) = A ± B
x→a
x→a
x→a
3. lim (f (x) · g(x)) = lim f (x) · lim g(x) = A · B
x→a
x→a
x→a
limx→a f (x)
A
f (x)
=
=
4. If B 6= 0, lim
x→a g(x)
limx→a g(x)
B
5. If p(x) is a polynomial then lim p(x) = p(a)
x→a
k
6. For any real number k, lim (f (x))k = lim f (x) , if the limit exists.
x→a
x→a
7. lim f (x) = lim g(x) if f (x) = g(x) for all x 6= a
x→a
x→a
8. For any real number b > 0, lim b
x→a
f (x)
=b
limx→a f (x)
9. For any real number b > 0, b 6= 1, lim (logb f (x)) = logb (lim f (x))
x→a
x→a
Limits as x approaches infinity
If the values f (x) can be made arbitrarily close to L by making x
sufficiently large in magnitude and positive (resp. negative), then we
write lim f (x) = L (resp. lim f (x) = L)
x→∞
x→−∞
Limits as x approaches infinity
If the values f (x) can be made arbitrarily close to L by making x
sufficiently large in magnitude and positive (resp. negative), then we
write lim f (x) = L (resp. lim f (x) = L)
x→∞
x→−∞
A thing to remember
1
1
For any positive real number n lim n = 0, lim+ n = ∞
x→∞ x
x→0 x
1
We also have lim n = 0 if this limit exists.
x→−∞ x
Limits as x approaches infinity
If the values f (x) can be made arbitrarily close to L by making x
sufficiently large in magnitude and positive (resp. negative), then we
write lim f (x) = L (resp. lim f (x) = L)
x→∞
x→−∞
A thing to remember
1
1
For any positive real number n lim n = 0, lim+ n = ∞
x→∞ x
x→0 x
1
We also have lim n = 0 if this limit exists.
x→−∞ x
x
e
Also, lim xn e−x = 0, lim n = ∞, lim+ xn ln x = 0
x→∞
x→∞ x
x→0
Limits as x approaches infinity
If the values f (x) can be made arbitrarily close to L by making x
sufficiently large in magnitude and positive (resp. negative), then we
write lim f (x) = L (resp. lim f (x) = L)
x→∞
x→−∞
A thing to remember
1
1
For any positive real number n lim n = 0, lim+ n = ∞
x→∞ x
x→0 x
1
We also have lim n = 0 if this limit exists.
x→−∞ x
x
e
Also, lim xn e−x = 0, lim n = ∞, lim+ xn ln x = 0
x→∞
x→∞ x
x→0
Change of variable
lim f (x) = lim f (t + a)
x→a
t→0
1
lim f (x) = lim f
t→∞
t
x→0+
1
lim f (x) = lim f
t→−∞
t
x→0−
Limits of rational functions
p(x)
, where p(x) and q(x) are polynomials.
Consider lim
x→∞ q(x)
Limits of rational functions
p(x)
, where p(x) and q(x) are polynomials.
Consider lim
x→∞ q(x)
To find the limit divide numerator and denominator by the largest power
of x in q(x) and then use rules for limits.
1 + 8/x2
limx→∞ (1 + 8/x2 )
1
x2 + 8
lim
= lim
=
=
2
x→∞ 6 − 1/x
x→∞ 6x − x
limx→∞ (6 − 1/x)
6
Limits of rational functions
p(x)
, where p(x) and q(x) are polynomials.
Consider lim
x→∞ q(x)
To find the limit divide numerator and denominator by the largest power
of x in q(x) and then use rules for limits.
1 + 8/x2
limx→∞ (1 + 8/x2 )
1
x2 + 8
lim
= lim
=
=
2
x→∞ 6 − 1/x
x→∞ 6x − x
limx→∞ (6 − 1/x)
6
p(x)
, where p(x) and q(x) are polynomials.
Consider lim
x→0 q(x)
Limits of rational functions
p(x)
, where p(x) and q(x) are polynomials.
Consider lim
x→∞ q(x)
To find the limit divide numerator and denominator by the largest power
of x in q(x) and then use rules for limits.
1 + 8/x2
limx→∞ (1 + 8/x2 )
1
x2 + 8
lim
= lim
=
=
2
x→∞ 6 − 1/x
x→∞ 6x − x
limx→∞ (6 − 1/x)
6
p(x)
, where p(x) and q(x) are polynomials.
Consider lim
x→0 q(x)
To find the limit divide numerator and denominator by the smallest
power of x in q(x) and then use rules for limits.
x+8
limx→0 (x + 8)
x2 + 8x
lim 2
= lim
=
= −8
x→0 6x − 1
x→0 6x − x
limx→0 (6x − 1)
Example
x2 − x − 6
(t − 2)2 − (t − 2) − 6
t2 − 5t
lim
= lim
= lim
= lim (t − 5) = −5
x→−2
t→0
t→0
t→0
x+2
(t − 2) + 2
t