SM015 - 8.2 (Notes Solutions)

8.2 Asymptotes

Friday, July 10, 2020 3:14 PM

Learning Outcomes
At the end of the lesson, students should be able to
(a) Find the vertical and horizontal asymptotes.

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Vertical Asymptotes

A line is a vertical asymptote of if either one of the

following one-sided limit obtained:

Horizontal Asymptotes

A line is a horizontal asymptote of if either one of the

following obtained:

8.2 Asymptotes Page 1

Example 1

Friday, May 29, 2020 10:48 PM

Find the vertical and horizontal asymptotes of
.

Hence, sketch the graph of .

Reference Graph: https://www.desmos.com/calculator/57vatrv0nh

Solution:

Note that is undefined for
For

Vertical Asymptote:

Horizontal Asymptote:

8.2 Asymptotes Page 2

Example 2 - ( Part I )

Friday, May 29, 2020 10:48 PM

Determine the asymptotes of .
Hence, sketch the graph of .

Reference Graph: https://www.desmos.com/calculator/8dgztlufa1

Solution:

Note that is undefined for
For

For

Vertical Asymptote:

Horizontal Asymptote:

8.2 Asymptotes Page 3

Example 2 - ( Part II )

Friday, May 29, 2020 10:48 PM

Hence, sketch the graph of .

Reference Graph: https://www.desmos.com/calculator/8dgztlufa1

Solution:

8.2 Asymptotes Page 4

Example 3 - ( Part I )

Friday, May 29, 2020 10:48 PM

Find the asymptotes of .
Hence, sketch the graph of .

Reference Graph: https://www.desmos.com/calculator/scbxfo822p

Solution:

Since

Note that is undefined for
For

For

Vertical Asymptote:

Horizontal Asymptote:

8.2 Asymptotes Page 5

Example 3 - ( Part II )

Friday, May 29, 2020 10:48 PM

Hence, sketch the graph of .

Reference Graph: https://www.desmos.com/calculator/scbxfo822p

Solution:

8.2 Asymptotes Page 6

Example 4

Friday, May 29, 2020 10:48 PM

Find the asymptotes of .
Hence, sketch the graph of .

Reference Graph: https://www.desmos.com/calculator/mfx1mpsatq

Solution: .
Note that domain for is
This implies that is undefined for

Vertical Asymptote:

Note that .

has NO horizontal asymptote.

8.2 Asymptotes Page 7

Example 5

Friday, May 29, 2020 10:48 PM

Find the asymptotes of .

Reference Graph: https://www.desmos.com/calculator/ohplneejqr

Solution: .
Note that

has NO vertical asymptote.

Horizontal Asymptote:

8.2 Asymptotes Page 8