# Worksheet: Existence of Limits

In this worksheet, we will practice determining whether the limit of a function at a certain value exists.

**Q3: **

Discuss the existence of given

- AThe limit does not exist because .
- BThe limit exists and equals 5.
- CThe limit does not exist because does not exist.
- DThe limit exists and equals .
- EThe limit does not exist because does not exist.

**Q4: **

Discuss the existence of given .

- AThe limit does not exist because does not exist.
- BThe limit exists and equals .
- CThe limit exists and equals .
- DThe limit does not exist because .
- EThe limit does not exist because does not exist.

**Q5: **

Given that find .

**Q6: **

Discuss the existence of given

- AThe limit exists and equals .
- BThe limit exists and equals .
- CThe limit does not exist because .
- DThe limit does not exist because does not exist.
- EThe limit does not exist because does not exist.

**Q7: **

Find given

**Q8: **

Discuss the existence of given

- A does not exist because .
- B exist and equals 3.
- C does not exist because exist, but does not exist.
- D does not exist because exist, but does not exist.

**Q11: **

Given that determine .

- A
- B
- C
- D5

**Q12: **

Given that , determine .

- A7
- B0
- CThe limit does not exist.
- D

**Q13: **

Given the function , what is true of ?

- AThe limit exists and equals 13.
- BThe limit does not exist because .
- CThe limit exists and equals .

**Q14: **

Determine .

**Q15: **

What can be said of for the function

- A exists and equals 9.
- B exists and equals .
- C does not exist because exists, but does not exist.
- D exists and equals .
- E does not exist because exists, but does not exist.

**Q16: **

Given that find , if it exists.

**Q17: **

Find the values of and given and exist where

- A,
- B,
- C,
- D,
- E,

**Q18: **

Suppose that What can be said about the existence of ?

- AThe limit exists and equals .
- BThe limit exists and equals .
- CThe limit exists and equals 0.
- DThe limit does not exist because .

**Q20: **

The figure shows the graph of the function

What is the -coordinate of point ? Give your answer as a fraction.

- A
- B
- C
- D
- E

What is the -coordinate of point ? Give your answer as a fraction.

- A
- B
- C
- D
- E

Setting the -coordinate of above as , give a formula of the sequence , , with .

- A
- B
- C
- D
- E

What does evaluation of with that was previously obtained tell you about a possible value for ?

What does evaluation of with tell you about a possible value for ?

What, therefore, can we conclude about ?

- AWe cannot conclude anything.
- B
- CWe conclude that .
- D
- EWe conclude that the one-sided limit does not exist.

What law of limits is used to prove that ?

- AThe limit of the quotient of functions is the quotient of the limits of the given functions if they exist and the denominator is nonzero.
- BThe limit of a difference of functions is the difference of the limits of the given functions if they exist.
- CThe limit of the product of functions is the product of the limits of the given functions if they exist.
- DThe limit of a sum of functions is the sum of the limits of the summand functions if they exist.
- EThe limit of a constant function as approaches any point is the constant itself.

What can be said about ?

- AIt does not exist because does not exist.
- BIt exists for some points on the number line and does not for other points.
- CIt exists and is sometimes 0.5 and sometimes 0.
- DIt exists and is equal to 0.5 because this is .
- EIt exists and is any value that you like.

**Q21: **

Discuss the existence of , where and

- A exists and equals 9.
- B exists and equals .
- C does not exist because exists, but does not exist.
- D does not exist because exists, but does not exist.
- E exists and equals 4.

**Q22: **

Determine .

- AThe limit does not exist.
- B10
- C3
- D0

**Q23: **

Discuss the existence of given

- AThe limit does not exist because.
- BThe limit exists and equals 24.
- CThe limit exists and equals 5.
- DThe limit exists and equals 20.
- EThe limit exists and equals 0.

**Q24: **

Given that determine .

- A1
- BThe limit does not exist.
- C0
- D

**Q25: **

Discuss the existence of given

- AThe limit exists and equals .
- BThe limit exists and equals .
- CThe limit exists and equals .
- DThe limit exists and equals .
- EThe limit does not exist because .