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Raising a new Generation of Leaders
MAT 121: LIMITS OF FUNCTIONS
IMAGA O. F.
INTRODUCTION
Find the value of the function
at x  1.
( x 2  1)
f ( x) 
x 1
At x  1 , the value of the function is 0 0 , which is indeterminate.
Suppose we decide to approach it closer and closer
x
0.5
0.9
0.99 0.999 0.9999 0.99999…
f(x) 1.50000 1.90000 1.99000 1.99900 1.99990 1.99999…
Now we see that as x gets closer to 1, f(x) gets close to 2.
2
Definition of a Limit
If the values of f(x) can be made as close as we like to a unique
number L by taking values of x sufficiently close to a (but not equal to
a), then we write
lim f ( x)  L
x a
which is read “the limit of f(x) as x approaches a is L”
Note that f(x) approaches L means the absolute difference between
f(x) and L i.e. |𝑓 𝑥 − 𝐿| can be made as small as we please.
3
Estimating a Limit Numerically
Use a table to estimate the following limits numerically.
1)
lim(3 x  2)
x 2
2)
2 x2  5x  2
lim
x2
x2
3)
x 1
lim
x1 x  1
4
Computation of Limits
Basic limits
Let b and c real numbers and let n be a positive integer
lim
3

3
1. lim
e.g.
,
.
b

b
lim
3

3
x 25
x c
x 
lim
x


2
x

c
lim
x

0
2. lim
e.g.
,
.
x 2
x c
x 0
2
2
lim
x

5

25
x

c
3. lim
e.g.
.
x 5
x c
n
n
lim
x

9

3
x

c
4. lim
for
n
even
and
c>0
e.g.
x 9
x c
n
n
5
Properties of Limits
Let b and c be real numbers, let n be a positive integer, and
let f and g be functions with the following limits
lim
f(x)  L
g( x )  K
and lim
x c
x c
1. Scalar multiple:
2. Sum or difference:
3. Product:
4. Quotient:
5. Power:
lim[
bf
(
x
)]

bL
x c
lim[
f
(
x
)

g
(
x
)]

L

K
x c
lim[
f
(
x
)
g
(
x
)]

LK
x c
f(x) L
lim

x c
g( x ) K
n
n
lim[
f
(
x
)]

L
x c
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1. Direct Substitution
To find the limit of f ( x ) as x  c we simply substitute c for x
f
(
x
)

f
(
c
)
i.e. lim
x c
This method is used for polynomial functions, rational
functions with nonzero denominators, trigonometric
functions.
E.g. Find the following limits
a. lim 𝑥 2 + 𝑥 − 6 = 12 + 1 − 6 = 2 − 6 = −4
𝑥→1
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a
x2  x 6
lim
x 1
x 3
c.
lim
( x cos x )
x 
e.
2𝑥 +23−𝑥 −6
lim −𝑥 2 1−𝑥
−2
𝑥→2 2
b.
lim
sin
x
x 
d.
lim
(
x

4)
x 3
2
8
2. Dividing Out Technique
It is used to find limits of rational functions f(x)/g(x) in which the
numerator and denominator both have a limit of zero as x
approaches c i.e. an indeterminate form of type 0/0.
Find the limit of the following functions:
a.
b.
c.
2
2x  8
x  3 x  10
x  6x  9
lim
lim
lim
2
2
x 4
x 5
x 3
x  x  12
x  10 x  25
x 3
3 2
x 1
𝑥 −2 3 𝑥+1
d. lim
e. lim
3
2
x 1
𝑥−1 2
𝑥→1
x  x  x 1
2
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3. Rationalization Technique
This is a method of finding the limit of a function by first by
first rationalizing either the numerator or denominator. To
rationalize is to multiply the function with the conjugate of
either the numerator or denominator. For instance the
conjugate of 4  1 is 4  1.
Find the limits of the following functions
x 1
x  1 1
a. lim
b. lim
c. lim
x 1
x 0
x 1
𝑥→−2
x
𝑥+10−3
𝑥+1
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One-Sided Limits
If the values of f(x) can be made close as we like to L by
taking values of x sufficiently close to a (but greater than a),
then we write
lim f ( x)  L
x a 
11
and if the values of f(x) can be made close as we like to L by
taking values of x sufficiently close to a (but less than a), then
we write
lim f ( x)  L
x a 
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Find the limit as 𝑥 → 0 from the left and the limit as 𝑥 → 0
from the right for
2x
f(x) 
x
Find the limit as 𝑥 → 2 from the left and the limit as 𝑥 → 2 from
the right for (i)
c.
Find lim+
𝑥→3
𝑓 𝑥 =
2
𝑥−2 𝑥 2 +𝑥
𝑥−2
𝑥−3+𝑥 +1
ii)
4 − 𝑥2
𝑥
d. lim−
𝑥→0 𝑥
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Existence of a Limit
If f is a function and c and L are real numbers, then
lim f ( x)  L
x c
if and only if both the left and right limits exists and are equal to L.
Discuss the existence of the limit of
4−𝑥
a. 𝑓 𝑥 =
4𝑥 − 𝑥 2
𝑥<1
as 𝑥 → 1.
𝑥>1
2𝑥 + 5
b. 𝑔(𝑥) = 3
𝑥 − 8𝑥 + 1
𝑥<3
as 𝑥 → 3
𝑥>3
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