Section 4.1 Worksheet
Definitions
To understand all the definitions of limits of functions, we need to understand the building
blocks of these definitions. Here is a dictionary of key concepts that we need to be able to
translate into precise mathematical language:
1. x is close but not equal to
(a) a 2 R from the right:
(b) a 2 R from the left:
(c) a 2 R:
(d) 1:
(e)
1:
a<x<a+
a
<x<a
0 < |x
a| <
x>m
x<m
x ! a+
x!a
x!a
x!1
x!
1
2. f (x) is close to
(a) L 2 R:
(b) 1:
(c)
1:
|f (x)
L| < ✏
f (x) > M
f (x) < M
limx!? f (x) = L
limx!? f (x) = 1
limx!? f (x) =
1
If you choose any one of the five options for x and any one of the three options for f (x), you
can build a definition of limit (15 definitions total!). You need to make sure f is defined on
an open interval in such a way that your choice for x makes sense. For instance:
Definition (1a, 2a). Let f be a function defined on an open interval I = (a, b) for a 2 R
and b 2 R [ {1}, and let L 2 R. Then limx!a+ f (x) = L if for every ✏ > 0, there is > 0
such that for all x 2 I satisfying a < x < a + , |f (x) L| < ✏.
Definition (1c, 2b). Let f be a function defined on an open interval I containing a 2 R.
Then limx!a f (x) = 1 if for every M 2 R, there is > 0 such that for all x 2 I satisfying
0 < |x a| < , f (x) > M .
Definition (1e, 2c). Let f be a function defined on an open interval I = ( 1, b) for some
b 2 R [ {1}. Then limx! 1 f (x) = 1 if for every M 2 R, there is m 2 R such that for
all x 2 I satisfying x < m, f (x) < M .
Exercise 1. Carefully state definitions for
(i) 1a, 2c
(ii) 1b, 2a
(iii) 1c, 2a
(iv) 1d, 2b
(v) 1e, 2b
1
Examples
Once we know how to construct the definitions, we need to be able to give examples of
functions that have those limits. For 1a-c and 2a, we should be able to give examples for
any choice of values for a and L.
Example. p• 1a, 2a with p
a = 0 and L = 6: Examples include f (x) = 6, f (x) = x + 6,
f (x) = x + 6, f (x) = x3 + 36, and f (x) = 6x
.
x
• 1c, 2b with a = 2: Examples include f (x) =
1
,
|x 2|
• 1e, 2c: Examples include f (x) = x, f (x) = 3x
f (x) =
1
,
(x 2)2
4, and f (x) =
and f (x) =
100
|x 2|
999.
x2 .
Exercise 2. Give at least three examples of functions for each of the limits in Exercise 1.
Try to choose simple examples!
Proofs
Finally, we need to practice proving that functions have certain limits using only our definitions. For this, we should choose relatively simple functions, since otherwise the proofs
could become really complicated.
Claim (1a, 2a). Let f (x) = x + 6. Then limx!0+ f (x) = 6.
= ✏. Then for all x 2 (0, 1) such that 0 < x < ,
Proof. Given ✏ > 0, choose
|f (x)
Claim (1c, 2b). Let f (x) =
6| = |x + 6
1
.
|x 2|
Proof. Given M 2 R, choose
f (x) =
Claim (1e, 2c). Let f (x) = 3x
Then for all x 2 R such that 0 < |x
1
|x
2|
>
1
Proof. Given M 2 R, choose m =
M +4
.
3
2| < ,
= |M | + 1 > M.
4. Then limx!
f (x) = 3x
= ✏.
Then limx!2 f (x) = 1.
1
.
|M |+1
=
6| = |x| = x <
1
f (x) =
1.
Then for all x 2 R such that x < m,
4 < 3m
4=M +4
4 = M.
Exercise 3. For each limit definition you constructed in Exercise 1, choose one of your
examples in Exercise 2 (use an easy one!) and prove that your chosen function has that limit
using only your definition.
2