3. •
•
•
Limits is our best prediction of a point we
didn’t observe.
Limits give us an estimate when we can’t
compute a result directly.
The limit wonders, “If you can see
everything except a single value, what do you
think is there?”.
11. FORMAL DEFINITION OF LIMIT
Let f(x) be any function
Let a and L be the numbers
If we can make f(x) as close to L as we please
by choosing x sufficiently close to a then we
say that the limit of f(x) as x approaches a is
L or symbolically,
lim �� �� = �
�
��→��
12. We will study how f(x) changes as the value of
x approaches c, in symbols �� → ��.
Examples:
1. Consider �� �� =
��2 + 5 as x → 2
x 1.84 1.98 2.03 2.08 2.19 2.55
f(x) 8.39 8.92 9.12 9.33 9.80 11.50
��2 + 5 → 9 ��
�� �� → 2
lim ��2 + 5
= 9
��→2
16. The limit of a function f(x) is equal to L
as x approaches a, written as
��
→�
�−
��
→�
�+
lim �� �� = �� if
and only if
��→��
lim �� �� = �
� = lim �� �
�
17. THEOREMS ON LIMITS
2
Let a and c be real numbers so
that
lim ��(��) and lim ��(��)
exist.
��→����→��
1. Constant Rule
Examples:
1.1 lim 8 = 8
��→��
1.2 lim 7.21 = 7.21
��→��
lim �� =
��
��→��
23. 6.2. If lim �� �� = 0 and lim
�� �� = −2
��→�� ��→��
lim
�
�(
�
�)
��→�
� ��
(��)
lim ��
(��)
�
� →
�
�
= ��→��
=
0
lim ��(��)
−2
= 0
6.3. If lim �� �� = 5 and
lim �� �� = 0
��→�� ��→��
DNE
not possible
to evaluate
24. 7. Power
Rule
If n is a positive integer and f(x)≥ 0 if
n is even, then lim[�� �� ]�� =
[lim ��(��)]��
��→�� ��→��
Examples:
7.1. If lim
�� ��
��→�
�
= 3, then
lim(�� �� )3 = (lim ��
�� )3 = 33 = 27
��→�� ��→��
36. Some limits cannot be evaluated simply using
direct substitution. Particularly in cases when
the given is a rational function, direct
substitution sometimes yields a number where
both the numerator and denominator are 0. The
expressio
n
0
0 is an example of an indeterminate
form.
42. ONE-SIDED LIMITS
3
•
One-sided limits are the same as
normal limits, we just restrict x so that
it approaches from just one side.
Right-hand Limit
Left-hand Limit
43. RIGHT-HAND LIMIT
Let f be a fu n c ti on def ined on s om e open
interval. Then the limit of the function f as x
approaches a from the right is L, which is
written as
lim �� �� = ��
��→��+
44. LEFT-HAND LIMIT
Let f be a fu n c ti on def ined on s om e open
interval. Then the limit of the function f as x
approac h es a from th e left is L , wh ich is
written as
lim �� �� = ��
��→��−
45. �
�
→
�
�
The lim ��(��) exists and is
equal to L iff
i. lim ��(��) and lim �
�(��) exists; and
��→��−
��→��
+
ii. lim ��(��) = lim ��(�
�) = L
��→��−
��→��
+
46. Example: The graph of a function f is shown
below. Use it to state the values (if they exist) of
the following:
a. lim �� �� b.
��→2− ��
→2+
lim
��
��
c. lim
�� �
�
��→2
��
→5−
d. lim �� �� e.
��
→5+
lim
��
��
f.lim �
� �
�
��→5
47. Example: The graph of a function g is shown
below. Use it to state the values (if they exist) of
the following:
a. lim �� �� b. lim
�� ��
��→1− ��→1+
c. lim
�� �
�
��→1
d. lim �� �� e. g(5)
��→5