Download The Plan 1.Properties of limits 2.Defined and undefined limits 3

The Plan
1.Properties of limits
2.Defined and undefined limits
3.Examples
1.Given two defined limits with functions 𝑓 & 𝑔,
constants 𝑐, 𝑘 ∈ ℝ and lim → 𝑓(𝑥) = 𝐿 and
lim → 𝑔(𝑥) = 𝑀
The following properties hold:
a. lim → 𝑘 =
b.lim
→
c. lim
→
d.lim
→
e. lim
→
𝑥=
𝑓(𝑥) ± 𝑔(𝑥) =
𝑘𝑓(𝑥) =
𝑓(𝑥) ∙ 𝑔(𝑥) =
f. lim
→
( )
( )
g.lim
→
𝑓(𝑥) =
h.lim
→
𝑓(𝑥)
=
=
2.Defined and undefined limits
If 𝑐 ∈ 𝒟 and 𝑓 is continuous then
lim → 𝑓(𝑥) = 𝑓(𝑐).
lim → 𝑓(𝑥) = 𝐿
iff lim → 𝑓(𝑥) exists and lim →
and both limits are equal to 𝐿.
𝑓(𝑥) exists
lim → 𝑓(𝑥) asks how 𝑓 behaves when 𝑥
becomes very large
lim → 𝑓(𝑥) asks how 𝑓 behaves when 𝑥
becomes very large in a negative direction
Sometimes limits are undefined, that is
lim → 𝑓(𝑥) ≠ 𝐿 for all 𝐿 ∈ ℝ.
A limit can be undefined because the numbers
become very large in a negative or positive
direction, in which case we say
lim → 𝑓(𝑥) = −∞ or lim → 𝑓(𝑥) = ∞.
A limit may be undefined because the left and
right limits differ.
lim → 𝑓(𝑥) = 𝐿 ≠ 𝐿 = lim → 𝑓(𝑥).
A limit may be undefined because the left or
right limit is undefined.
A limit may not exist because the function is
discontinuous in an infinite number of places
near 𝑐 or is otherwise very strange. (We do not
have to worry about these sorts of functions in
M1510)
3.Examples:
a. Find the limits or state why they do not
exist.
i. lim → 𝑥 =
ii. lim
→
𝑥 =
iii. lim
→
|𝑥| =
iv. lim
→
v. lim
→
=
= 𝑥
vi. 𝑔(𝑥) = 0
−𝑥
vii. lim
→
=
𝑥<3
𝑥 = 3 lim 𝑔(𝑥)
→
𝑥>3
viii. lim
→
ix. lim
→
x. lim
→
xi. lim
→
xii. lim
→
xiii. lim
→
xiv. lim
→
=
| |
=
| |
=
(3𝑥 − 2𝑥 + 1) =
= =
=
b.Sketch a function that meets the given
criteria.
i. lim → 𝑓(𝑥) = 4, lim → 𝑓 (𝑥) = 2
lim → 𝑓(𝑥) = 2, 𝑓 (3) = 3, and
𝑓(−2) = 1
ii. lim → 𝑓(𝑥) = 1, lim → 𝑓(𝑥) = −1,
lim → 𝑓(𝑥) = 0, lim → 𝑓(𝑥) = 1,
𝑓(2) = 1, and 𝑓(0) is undefined.