Worksheet: Limits and Asymptotic Behavior

In this worksheet, we will practice using limits to understand asymptotic behavior of functions.

Q1:

The function 𝑓(𝑥)=3𝑥33𝑥+80(𝑥3)(𝑥5)(𝑥7) is found to have a vertical asymptote at 𝑥=3. To which of {,+} do the values of 𝑓(𝑥) go as 𝑥 approaches 3 from the left and the right?

  • A+,
  • B,+
  • C+,+
  • D,

Q2:

Find the horizontal and vertical asymptotes of the function 𝑓(𝑥)=2𝑥+2𝑥𝑥𝑥sincossincos.

  • AHorizontal: does not exist, vertical: 𝑥=𝜋𝑛
  • BHorizontal: 𝑦=𝜋3, vertical: 𝑥=𝜋2+𝜋𝑛
  • CHorizontal: does not exist, vertical: 𝑥=𝜋4+𝜋𝑛
  • DThere are no vertical or horizontal asymptotes.
  • EHorizontal: 𝑦=𝜋, vertical: does not exist

Q3:

Find the horizontal and vertical asymptotes of 𝑓(𝑥)=2𝑥+2𝑥1.

  • AHorizontal: 𝑦=2, vertical: 𝑥=1
  • BHorizontal: 𝑦=1, vertical: 𝑥=1
  • CHorizontal: 𝑦=2, vertical: 𝑥=1
  • DHorizontal: 𝑦=2, vertical: 𝑥=1
  • EHorizontal: 𝑦=1, vertical: 𝑥=2

Q4:

For each of the following functions 𝑓, determine the horizontal asymptote(s) for 𝑓.

𝑓(𝑥)=52𝑥

  • A𝑦=3
  • B𝑥=3
  • C𝑦=5
  • D𝑥=5
  • E𝑦=2

𝑓(𝑥)=𝑥𝑥sin

  • A𝑥=5
  • B𝑦=1
  • C𝑥=4
  • D𝑦=0
  • E𝑦=1

Q5:

Find the horizontal and vertical asymptotes of 𝑓(𝑥)=𝑥𝑥+𝑥.

  • AHorizontal: 𝑦=1, vertical: 𝑥=1
  • BHorizontal: 𝑦=1, vertical: 𝑥=0
  • CHorizontal: 𝑦=1, vertical: 𝑥=0
  • DHorizontal: 𝑦=0, vertical: 𝑥=0 and 𝑥=1
  • EHorizontal: 𝑦=0, vertical: 𝑥=1

Q6:

Find the horizontal and vertical asymptotes of the function 𝑓(𝑥)=3𝑥3𝑥.

  • AHorizontal: 𝑦=1, vertical: 𝑥=1
  • BHorizontal: none, vertical: 𝑥=0
  • CHorizontal: none, vertical: 𝑥=1
  • DHorizontal: 𝑦=1, vertical: none
  • EHorizontal: 𝑦=0, vertical: none

Q7:

Find the horizontal and vertical asymptotes of 𝑓(𝑥)=1𝑥12.

  • AHorizontal: 𝑦=2, vertical 𝑥=1
  • BHorizontal: 𝑦=2, vertical 𝑥=0
  • CHorizontal: 𝑦=1, vertical 𝑥=2
  • DHorizontal: 𝑦=2𝑥, vertical 𝑥=1
  • EHorizontal: 𝑦=2, vertical 𝑥=0

Q8:

Find the horizontal and vertical asymptotes of the function 𝑓(𝑥)=3𝑥𝑥sin.

  • AHorizontal: 𝑦=1, vertical: 𝑥=2
  • BThere are no vertical or horizontal asymptotes.
  • CHorizontal: 𝑥=1, vertical: does not exist
  • DHorizontal: does not exist, vertical: 𝑥=1
  • EHorizontal: 𝑦=1, vertical: 𝑥=2

Q9:

Find the horizontal and vertical asymptotes of 𝑓(𝑥)=𝑥(𝑥)𝑥1sin.

  • AHorizontal: 𝑦=0, vertical: 𝑥=1, 𝑥=0
  • BHorizontal: 𝑦=0, vertical: 𝑥=1, 𝑥=1
  • CHorizontal: 𝑦=1, vertical: 𝑥=1, 𝑥=1
  • DHorizontal: 𝑦=0, vertical: 𝑥=1
  • EHorizontal: 𝑦=1, vertical: 𝑥=0

Q10:

Find the horizontal and vertical asymptotes of the function 𝑓(𝑥)=43𝑥.

  • AHorizontal: 𝑦=4, vertical: 𝑥=0
  • BHorizontal: 𝑦=4, vertical: 𝑥=4
  • CHorizontal: 𝑦=0, vertical: 𝑥=0
  • DHorizontal: 𝑦=0, vertical: 𝑥=4
  • EHorizontal: 𝑦=𝑥, vertical: 𝑥=3

Q11:

Find the vertical asymptote of the function 𝑓(𝑥)=(𝑥+5)ln.

  • A𝑥=
  • B𝑥=15
  • C𝑥=5
  • D𝑥=15
  • E𝑥=5

Q12:

Find all the vertical asymptotes for the function 𝑓(𝑥)=5(𝑥3)(3𝑥+9𝑥30).

  • A𝑥=5, 𝑥=2, and 𝑥=3
  • B𝑥=0
  • C𝑥=3, 𝑥=2, and 𝑥=5
  • D𝑥=3
  • E𝑥=3, 𝑥=2, and 𝑥=5

Q13:

Find the vertical asymptotes of the function 𝑓(𝑥)=1(𝑥)+3(𝑥)cossin.

  • A𝑥=𝜋6+𝑛𝜋
  • B𝑥=𝜋6+𝑛𝜋
  • C𝑥=𝜋3+𝑛𝜋2
  • D𝑥=𝜋3+𝑛𝜋
  • E𝑥=𝜋6+𝑛𝜋2

Q14:

Find the horizontal and vertical asymptotes of the function 𝑓(𝑥)=23𝑥5+7.

  • AHorizontal:vertical:𝑦=53,𝑥=7
  • BHorizontal:vertical:𝑦=7,𝑥=53
  • CHorizontal:vertical:𝑦=7,𝑥=53
  • DHorizontal:vertical:𝑦=53,𝑥=7
  • EHorizontal:vertical:𝑦=0,𝑥=23

Q15:

Find the horizontal asymptotes of the function 𝑓(𝑥)=3𝑥+75𝑥4.

  • A𝑦=74
  • B𝑦=+25, 𝑦=25
  • C𝑦=35
  • D𝑦=35
  • E𝑦=10

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