Worksheet: Limits and Asymptotic Behavior

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

Q1:

The function 𝑓 ( 𝑥 ) = 3 𝑥 3 3 𝑥 + 8 0 ( 𝑥 3 ) ( 𝑥 5 ) ( 𝑥 7 ) 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 ) = 2 𝑥 + 2 𝑥 𝑥 𝑥 s i n c o s s i n c o s .

  • AHorizontal: 𝑦 = 𝜋 , vertical: does not exist
  • BHorizontal: 𝑦 = 𝜋 3 , vertical: 𝑥 = 𝜋 2 + 𝜋 𝑛
  • CHorizontal: does not exist, vertical: 𝑥 = 𝜋 4 + 𝜋 𝑛
  • DThere are no vertical or horizontal asymptotes.
  • EHorizontal: does not exist, vertical: 𝑥 = 𝜋 𝑛

Q3:

Find the horizontal and vertical asymptotes of 𝑓 ( 𝑥 ) = 2 𝑥 + 2 𝑥 1 .

  • AHorizontal: 𝑦 = 2 , vertical: 𝑥 = 1
  • BHorizontal: 𝑦 = 2 , vertical: 𝑥 = 1
  • CHorizontal: 𝑦 = 2 , vertical: 𝑥 = 1
  • DHorizontal: 𝑦 = 1 , 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 ) = 5 2 𝑥

  • A 𝑥 = 5
  • B 𝑦 = 2
  • C 𝑥 = 3
  • D 𝑦 = 5
  • E 𝑦 = 3

𝑓 ( 𝑥 ) = 𝑥 𝑥 s i n

  • A 𝑥 = 5
  • B 𝑦 = 1
  • C 𝑦 = 1
  • D 𝑦 = 0
  • E 𝑥 = 4

Q5:

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

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

Q6:

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

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

Q7:

Find the horizontal and vertical asymptotes of 𝑓 ( 𝑥 ) = 1 𝑥 1 2 .

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

Q8:

Find the horizontal and vertical asymptotes of the function 𝑓 ( 𝑥 ) = 3 𝑥 𝑥 s i n .

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

Q9:

Find the horizontal and vertical asymptotes of 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 ) 𝑥 1 s i n .

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

Q10:

Find the horizontal and vertical asymptotes of the function 𝑓 ( 𝑥 ) = 4 3 𝑥 .

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

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