1.2 Evaluating Limits
and One-Sided Limits
Evaluating Limits
We have already studied the three ways to evaluate limits:
Graphically
Numerically (with a table)
Algebraically
Today we will explore algebraically evaluating limits a little more
Evaluating Limits Algebraically
A. lim𝑥→3
[𝑥2 2 − 𝑥 ]
Example 1: Determine the limits algebraically.
Evaluating Limits Algebraically
B. lim𝑥→2
𝑥2+2𝑥+4
𝑥+2
Example 1: Determine the limits algebraically.
Evaluating Limits Algebraically
A. lim𝑡→2
𝑡2−3𝑡+2
𝑡2−4
Example 2: Determine the limits algebraically.
Evaluating Limits Algebraically
B. lim𝑥→0
1
2+𝑥−1
2
𝑥
Example 2: Determine the limits algebraically.
Evaluating Limits AlgebraicallyExample 2: Determine the limits algebraically.
C. lim𝑥→9
𝑥−3
𝑥−9
Evaluating Limits Algebraically
You need to “adjust” a function when the original substitution gives
you 0
0; this is called indeterminate form
Note that something like 2
0or −
5
0are different; use graphs or tables to
determine those limits
“Adjusting” can mean expanding and simplifying, factoring and
cancelling, combining multiple fractions into one, or multiplying by
the conjugate
One-Sided and Two-Sided Limits
For a limit to exist, the function must approach the same
value from both sides. Or,
lim lim limx c x c x c
f x f x f x
Example 3
A. Find
B. Does the
exist?
2
limx
f x
2
limx
f x
x
y
Example 4
Given c = 2,
A. Draw the graph of f.
B. Determine and
C. Does exist? If so, what is it? If not, explain.
3 , 2
1, 22
x x
f x xx
limx c
f x
lim .x c
f x
limx c
f x
x
y
Evaluating Limits, Example 5
Non-existing Limits
There are 3 reasons for a non-existing limit
1. The left and right limits do not match (or one does not exist)
2. Unbounded behavior (aka an asymptote)
3. Oscillating behavior
Non-existing Limits, Example 6
Algebraically, how do we know that these do not have limits?
Non-existing Limits, Example 7
First, look at the graph or a table for the functions.
Algebraically, how do we know that these do not have limits?
A. B. Exception**
https://sites.math.washington.edu/~conroy/general/sin1overx/
0
1limsinx x
0
1lim sinx
xx
In Summary…
There are 3 ways to find a limit:
Numerically (a table)
Graphically
Algebraically (substitution, adjusting)
In order for a limit to exist, the left and right limits must be the same
The function does not have to exist at the c value, nor does the
function value have to equal the limit in order for the limit to exist
There are 3 times a limit will not exist
One-sided limits are not equal/do not exist
Unbounded behavior
Oscillating behavior
Homework!
P. 66 #15 – 27 odd, 37, 39, 49, 51, 63