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.

  • Alim𝑓(𝑥)=0
  • B𝑓(6)=0
  • C𝑓(0)=6
  • Dlim𝑓(𝑥)=6

Q2:

What does the notation lim𝑓(𝑥)=5 mean?

  • AAs 𝑥 gets closer and closer to 5 , 𝑓(𝑥) gets closer and closer to 2 .
  • 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.
  • DThe value of the function 𝑓 is 2 at 𝑥=5.
  • E𝑓 is undefined at 𝑥=2 and at 𝑥=5.

Q3:

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

𝑥2.992.9992.99992.99993.000013.00013.0013.01
𝑓(𝑥)5.895.915.975.996.00026.0076.016.02

What does the table suggest about the value of lim𝑓(𝑥)?

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

Q4:

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

𝑥4.8954.9024.9794.99995.00015.00045.0075.02
𝑓(𝑥)14.0114.00514.00314.000113.999913.99813.9213.895

What does the table suggest about the value of lim𝑓(𝑥)?

  • AThe value of the limit does not exist.
  • BThe value of the limit equals 14.
  • CThe value of the limit equals 13.
  • DThe value of the limit equals 5.
  • EThe value of the limit equals 4.

Q5:

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

  • AFalse
  • BTrue

Q6:

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

  • AFalse
  • BTrue

Q7:

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

What does the graph suggest about the value of lim𝑓(𝑥)?

  • Alim𝑓(𝑥)=2
  • Blim𝑓(𝑥)=2
  • Clim𝑓(𝑥)=4
  • Dlim𝑓(𝑥) does not exist.
  • Elim𝑓(𝑥)=0

Q8:

The following figure represents the graph of the function 𝑓.

What does the graph suggest about the value of lim𝑓(𝑥)?

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

Q9:

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

What is the value of 𝑓(0)?

  • A𝑓(0) is undefined.
  • B𝑓(0)=3.1
  • C𝑓(0)=1
  • D𝑓(0)=0
  • E𝑓(0)=3.1

What does the graph suggest about the value of lim𝑓(𝑥)?

  • Alim𝑓(𝑥) does not exist.
  • Blim𝑓(𝑥)=3.1
  • Clim𝑓(𝑥)=3.1
  • Dlim𝑓(𝑥)=1
  • Elim𝑓(𝑥)=0

Q10:

The following figure is the graph of the function 𝑓, where 𝑓(𝑥)=4(𝑥5)𝑥1(𝑥1)(𝑥5)𝑥5.

What is the value of 𝑓(5)?

  • A𝑓(5)=0
  • B𝑓(5) is undefined.
  • C𝑓(5)=5
  • D𝑓(5)=4
  • E𝑓(5)=3

What does the graph suggest about the value of lim𝑓(𝑥)?

  • Alim𝑓(𝑥)=3
  • Blim𝑓(𝑥)=0
  • Clim𝑓(𝑥)=4
  • Dlim𝑓(𝑥)=5
  • Elim𝑓(𝑥) does not exist.

Q11:

Which of the following statements is not the same as saying that lim𝑓(𝑥)=3?

  • A𝑓(3) is equal to 𝑓(8).
  • B𝑓(𝑥) approaches 3 as 𝑥 approaches 8 .
  • CWe can make 𝑓(𝑥) as close as we like from 3 by taking 𝑥 sufficiently close to 8 .
  • DAs 𝑥 gets closer and closer to 8 , 𝑓(𝑥) gets closer and closer to 3 .

Q12:

If 𝑓(6)=6, what can we say about lim𝑓(𝑥)?

  • Alim𝑓(𝑥)6
  • Blim𝑓(𝑥)=0
  • Clim𝑓(𝑥)=6
  • DWe cannot draw any conclusions about lim𝑓(𝑥).
  • Elim𝑓(𝑥)=1

Q13:

Given that lim𝑔(𝑥)=10, what is the value of lim𝑔(𝑥)?

  • A10
  • B20
  • C7
  • D5
  • E13

Q14:

Given that lim𝑓(𝑥)=6, which of the following statements must be false?

  • A𝑓(2) is undefined.
  • Blim𝑓(𝑥)=6
  • C𝑓(2)=6
  • Dlim𝑓(𝑥)=4
  • E𝑓(2)=4

Q15:

Given that lim𝑓(𝑥)=4, which of the following statements must be true?

  • A𝑓(4)=1
  • B𝑓(1)=4
  • C𝑓(1)4
  • D𝑓(4)1
  • ENone of the above

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