Worksheet: Limits and Limit Notation

In this worksheet, we will practice using limit notation and exploring the concept of a limit.

Q1:

What is the correct notation that describes the following statement?

As 𝑥 approaches 0 , 𝑓 ( 𝑥 ) approaches 6 .

  • A 𝑓 ( 0 ) = 6
  • B l i m 𝑓 ( 𝑥 ) = 0
  • C 𝑓 ( 6 ) = 0
  • D l i m 𝑓 ( 𝑥 ) = 6

Q2:

What does the notation l i m 𝑓 ( 𝑥 ) = 5 mean?

  • A 𝑓 is undefined at 𝑥 = 2 and at 𝑥 = 5 .
  • BAs 𝑥 gets closer and closer to 2 , 𝑓 ( 𝑥 ) gets closer and closer to 5 .
  • CThe value of 𝑓 at 𝑥 = 2 is equal to the value of 𝑓 at 𝑥 = 5 .
  • DAs 𝑥 gets closer and closer to 5 , 𝑓 ( 𝑥 ) gets closer and closer to 2 .
  • EThe value of the function 𝑓 is 2 at 𝑥 = 5 .

Q3:

The following table gives the values of the function 𝑓 at several values of 𝑥 .

𝑥 2.99 2.999 2.9999 2.9999 3.00001 3.0001 3.001 3.01
𝑓 ( 𝑥 ) 5 . 8 9 5 . 9 1 5 . 9 7 5 . 9 9 6 . 0 0 0 2 6 . 0 0 7 6 . 0 1 6 . 0 2

What does the table suggest about the value of l i m 𝑓 ( 𝑥 ) ?

  • AThe value of the limit equals 2.
  • BThe value of the limit equals 3.
  • CThe value of the limit equals 6 .
  • DThe value of the limit equals 5 .

Q4:

The following table gives the values of the function 𝑓 at several values of 𝑥 .

𝑥 4 . 8 9 5 4 . 9 0 2 4 . 9 7 9 4 . 9 9 9 9 5 . 0 0 0 1 5 . 0 0 0 4 5 . 0 0 7 5 . 0 2
𝑓 ( 𝑥 ) 1 4 . 0 1 1 4 . 0 0 5 1 4 . 0 0 3 1 4 . 0 0 0 1 1 3 . 9 9 9 9 1 3 . 9 9 8 1 3 . 9 2 1 3 . 8 9 5

What does the table suggest about the value of l i m 𝑓 ( 𝑥 ) ?

  • AThe value of the limit equals 5 .
  • BThe value of the limit equals 4 .
  • CThe value of the limit equals 1 3 .
  • DThe value of the limit equals 1 4 .
  • EThe value of the limit does not exist.

Q5:

True or False: If l i m 𝑓 ( 𝑥 ) = 4 , then it is possible that the function 𝑓 could be undefined at 𝑥 = 1 .

  • AFalse
  • BTrue

Q6:

True or False: If l i m 𝑓 ( 𝑥 ) = 3 , then 𝑓 ( 5 ) must be equal to 3 .

  • ATrue
  • BFalse

Q7:

The following figure represents the graph of the function 𝑓 ( 𝑥 ) = 𝑥 .

What does the graph suggest about the value of l i m 𝑓 ( 𝑥 ) ?

  • A l i m 𝑓 ( 𝑥 ) = 2
  • B l i m 𝑓 ( 𝑥 ) does not exist.
  • C l i m 𝑓 ( 𝑥 ) = 4
  • D l i m 𝑓 ( 𝑥 ) = 0
  • E l i m 𝑓 ( 𝑥 ) = 2

Q8:

The following figure represents the graph of the function 𝑓 .

What does the graph suggest about the value of l i m 𝑓 ( 𝑥 ) ?

  • AThe value of the limit equals 4 .
  • BThe value of the limit equals 3.
  • CThe limit does not exist.
  • DThe value of the limit equals 3 .
  • EThe value of the limit equals 2.

Q9:

The following figure is the graph of the function 𝑓 , where 𝑓 ( 𝑥 ) = 𝑥 𝑥 s i n .

What is the value of 𝑓 ( 0 ) ?

  • A 𝑓 ( 0 ) = 0
  • B 𝑓 ( 0 ) = 3 . 1
  • C 𝑓 ( 0 ) = 3 . 1
  • D 𝑓 ( 0 ) = 1
  • E 𝑓 ( 0 ) is undefined.

What does the graph suggest about the value of l i m 𝑓 ( 𝑥 ) ?

  • A l i m 𝑓 ( 𝑥 ) = 3 . 1
  • B l i m 𝑓 ( 𝑥 ) = 0
  • C l i m 𝑓 ( 𝑥 ) = 3 . 1
  • D l i m 𝑓 ( 𝑥 ) does not exist.
  • E l i m 𝑓 ( 𝑥 ) = 1

Q10:

The following figure is the graph of the function 𝑓 , where 𝑓 ( 𝑥 ) = 4 ( 𝑥 5 ) 𝑥 1 ( 𝑥 1 ) ( 𝑥 5 ) 𝑥 5 .

What is the value of 𝑓 ( 5 ) ?

  • A 𝑓 ( 5 ) = 5
  • B 𝑓 ( 5 ) is undefined.
  • C 𝑓 ( 5 ) = 0
  • D 𝑓 ( 5 ) = 3
  • E 𝑓 ( 5 ) = 4

What does the graph suggest about the value of l i m 𝑓 ( 𝑥 ) ?

  • A l i m 𝑓 ( 𝑥 ) = 3
  • B l i m 𝑓 ( 𝑥 ) = 5
  • C l i m 𝑓 ( 𝑥 ) = 0
  • D l i m 𝑓 ( 𝑥 ) does not exist.
  • E l i m 𝑓 ( 𝑥 ) = 4

Q11:

Which of the following statements is not the same as saying that l i m 𝑓 ( 𝑥 ) = 3 ?

  • AAs 𝑥 gets closer and closer to 8 , 𝑓 ( 𝑥 ) gets closer and closer to 3 .
  • BWe can make 𝑓 ( 𝑥 ) as close as we like from 3 by taking 𝑥 sufficiently close to 8 .
  • C 𝑓 ( 𝑥 ) approaches 3 as 𝑥 approaches 8 .
  • D 𝑓 ( 3 ) is equal to 𝑓 ( 8 ) .

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