1.2 Evaluating Limits and One-Sided Limits
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# Theorem 3 One-Sided and Two-Sided Limits · 2018. 9. 1. · Evaluating Limits Algebraically B. lim 𝑥→2 𝑥2+2𝑥+4 𝑥+2 Example 1: Determine the limits algebraically.

Jan 01, 2021

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• 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

https://sites.math.washington.edu/~conroy/general/sin1overx/

• In Summary…

There are 3 ways to find a limit:

Numerically (a table)

Graphically

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

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