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Worksheet: Existence of Limits

In this worksheet, we will practice finding a limit and discussing the existence of a limit.

Q1:

Discuss the existence of l i m π‘₯ β†’ βˆ’ 1 5 𝑓 ( π‘₯ ) given

  • AThe limit does not exist because l i m π‘₯ β†’ βˆ’ 1 5 + 𝑓 ( π‘₯ ) exists, but l i m π‘₯ β†’ βˆ’ 1 5 βˆ’ 𝑓 ( π‘₯ ) does not exist.
  • BThe limit does not exist because l i m π‘₯ β†’ βˆ’ 1 5 βˆ’ 𝑓 ( π‘₯ ) exists, but l i m π‘₯ β†’ βˆ’ 1 5 + 𝑓 ( π‘₯ ) does not exist.
  • CThe limit does not exist because both l i m π‘₯ β†’ βˆ’ 1 5 βˆ’ 𝑓 ( π‘₯ ) and l i m π‘₯ β†’ βˆ’ 1 5 + 𝑓 ( π‘₯ ) exist, but are not equal.
  • DThe limit exists and equals 210.
  • EThe limit exists and equals βˆ’ 1 5 .

Q2:

Given that find l i m π‘₯ β†’ πœ‹ 2 𝑓 ( π‘₯ ) , if it exists.

  • A βˆ’ 3
  • B1
  • C βˆ’ 4
  • Ddoes not exist
  • E βˆ’ 7

Q3:

Suppose that What can be said about the existence of l i m π‘₯ β†’ βˆ’ 1 𝑓 ( π‘₯ ) ?

  • A The limit exists and equals 1.
  • B The limit does not exist because l i m l i m π‘₯ β†’ βˆ’ 1 π‘₯ β†’ βˆ’ 1 βˆ’ + 𝑓 ( π‘₯ ) β‰  𝑓 ( π‘₯ ) .
  • C The limit exists and equals 0.
  • D The limit exists and equals 1 5 .

Q4:

Discuss the existence of l i m  β†’  𝑓 ( π‘₯ ) given

  • AThe limit exists and equals 0.
  • BThe limit does not exist because l i m l i m  β†’   β†’  οŽͺ  𝑓 ( π‘₯ ) β‰  𝑓 ( π‘₯ ) .
  • CThe limit exists and equals πœ‹ 3 .
  • DThe limit exists and equals 1.
  • EThe limit exists and equals βˆ’ πœ‹ 3 .

Q5:

Discuss the existence of l i m  β†’   𝑓 ( π‘₯ ) given

  • AThe limit exists and equals βˆ’ 1 0 .
  • BThe limit exists and equals βˆ’ 9 .
  • CThe limit does not exist because l i m l i m  β†’    β†’   οŽͺ  𝑓 ( π‘₯ ) β‰  𝑓 ( π‘₯ ) .
  • DThe limit exists and equals 1.

Q6:

Discuss the existence of l i m π‘₯ β†’ 0 𝑓 ( π‘₯ ) given

  • AThe limit exists and equals 5 .
  • BThe limit exists and equals 1 0 .
  • CThe limit exists and equals 0 .
  • DThe limit does not exist because l i m l i m π‘₯ β†’ 0 π‘₯ β†’ 0 βˆ’ + 𝑓 ( π‘₯ ) β‰  𝑓 ( π‘₯ ) .

Q7:

Discuss the existence of l i m  β†’  ο‘½  𝑓 ( π‘₯ ) given

  • AThe limit exists and equals 9.
  • BThe limit exists and equals 0.
  • CThe limit exists and equals 1 9 .
  • DThe limit does not exist.

Q8:

Given that find l i m π‘₯ β†’ 0 𝑓 ( π‘₯ ) .

  • A 2 3
  • B5
  • C 3 2
  • D1

Q9:

Discuss the existence of l i m π‘₯ β†’ βˆ’ 1 𝑓 ( π‘₯ ) given

  • AThe limit exists and equals βˆ’ 2 0 .
  • BThe limit exists and equals 20.
  • CThe limit does not exist because 𝑓 ( βˆ’ 1 ) βˆ’ does not exist.
  • DThe limit does not exist because 𝑓 ( βˆ’ 1 ) β‰  𝑓 ( βˆ’ 1 ) βˆ’ + .

Q10:

Discuss the existence of l i m π‘₯ β†’ 0 𝑓 ( π‘₯ ) given

  • A l i m π‘₯ β†’ 0 𝑓 ( π‘₯ ) does not exist because l i m π‘₯ β†’ 0 βˆ’ 𝑓 ( π‘₯ ) exist, but l i m π‘₯ β†’ 0 + 𝑓 ( π‘₯ ) does not exist.
  • B l i m π‘₯ β†’ 0 𝑓 ( π‘₯ ) does not exist because l i m l i m π‘₯ β†’ 0 π‘₯ β†’ 0 βˆ’ + 𝑓 ( π‘₯ ) β‰  𝑓 ( π‘₯ ) .
  • C l i m π‘₯ β†’ 0 𝑓 ( π‘₯ ) does not exist because l i m π‘₯ β†’ 0 + 𝑓 ( π‘₯ ) exist, but l i m π‘₯ β†’ 0 βˆ’ 𝑓 ( π‘₯ ) does not exist.
  • D l i m π‘₯ β†’ 0 𝑓 ( π‘₯ ) exist and equals 3.

Q11:

Discuss the existence of l i m π‘₯ β†’ 4 𝑓 ( π‘₯ ) given

  • AThe limit exists and equals 1 3 9 .
  • BThe limit exists and equals 5.
  • CThe limit does not exist because l i m π‘₯ β†’ 4 βˆ’ 𝑓 ( π‘₯ ) does not exist.
  • DThe limit does not exist because l i m l i m π‘₯ β†’ 4 π‘₯ β†’ 4 βˆ’ + 𝑓 ( π‘₯ ) β‰  𝑓 ( π‘₯ ) .
  • EThe limit does not exist because l i m π‘₯ β†’ 4 + 𝑓 ( π‘₯ ) does not exist.

Q12:

Suppose that What can be said about the existence of l i m π‘₯ β†’ 0 𝑓 ( π‘₯ ) .

  • A The limit exists and equals βˆ’ 1 2 .
  • B The limit does not exist because l i m l i m π‘₯ β†’ 0 π‘₯ β†’ 0 βˆ’ + 𝑓 ( π‘₯ ) β‰  𝑓 ( π‘₯ ) .
  • C The limit exists and equals 5.
  • D The limit exists and equals βˆ’ 5 2 .

Q13:

Describe the limit as π‘₯ β†’ 0 for the following function:

  • AThe limit exists and equals βˆ’ 7 5 .
  • BThe limit exists and equals 7 5 .
  • CThe limit exists and equals 8 7 .
  • DThe limit does not exist because l i m l i m π‘₯ β†’ 0 π‘₯ β†’ 0 βˆ’ + 𝑓 ( π‘₯ ) β‰  𝑓 ( π‘₯ ) .
  • EThe limit exists and equals βˆ’ 8 7 .

Q14:

Discuss the existence of the limit as π‘₯ β†’ πœ‹ given

  • AThe limit exists and equals βˆ’ 9 .
  • BThe limit does not exist because l i m l i m π‘₯ β†’ πœ‹ π‘₯ β†’ πœ‹ βˆ’ + 𝑓 ( π‘₯ ) β‰  𝑓 ( π‘₯ ) .
  • CThe limit exists and equals βˆ’ 1 8 .
  • DThe limit exists and equals 9 .

Q15:

Determine l i m π‘₯ β†’ 0 𝑓 ( π‘₯ ) if it exists, where

  • A 1 5
  • B1
  • C4
  • D5

Q16:

Determine l i m π‘₯ β†’ 9 𝑓 ( π‘₯ ) , where

Q17:

Discuss the existence of l i m π‘₯ β†’ βˆ’ 6 𝑓 ( π‘₯ ) given 𝑓 ( π‘₯ ) = π‘₯ | π‘₯ + 7 | βˆ’ 3 .

  • A The limit does not exist because l i m l i m π‘₯ β†’ βˆ’ 6 π‘₯ β†’ βˆ’ 6 βˆ’ + 𝑓 ( π‘₯ ) β‰  𝑓 ( π‘₯ ) .
  • B The limit exists and equals βˆ’ 3 .
  • C The limit does not exist because l i m π‘₯ β†’ βˆ’ 6 βˆ’ 𝑓 ( π‘₯ ) does not exist.
  • D The limit exists and equals βˆ’ 9 .
  • E The limit does not exist because l i m π‘₯ β†’ βˆ’ 6 + 𝑓 ( π‘₯ ) does not exist.

Q18:

Given that determine l i m π‘₯ β†’ 1 𝑓 ( π‘₯ ) .

  • A πœ‹ 2
  • B2
  • C 1 2
  • D 2 πœ‹

Q19:

Given that and l i m π‘₯ β†’ 0 𝑓 ( π‘₯ ) exists, determine the possible values of π‘Ž .

  • A 2 5 4 , βˆ’ 2 5 4
  • B 2 5 , βˆ’ 2 5
  • C 4 2 5 , βˆ’ 4 2 5
  • D 5 2 , βˆ’ 5 2

Q20:

Given that the function has a limit when π‘₯ = 2 , determine the values of π‘Ž and 𝑏 .

  • A π‘Ž = βˆ’ 1 2 , 𝑏 = 2 8
  • B π‘Ž = βˆ’ 8 , 𝑏 = βˆ’ 1 2
  • C π‘Ž = βˆ’ 1 6 , 𝑏 = 3 6
  • D π‘Ž = βˆ’ 1 6 , 𝑏 = 2 8

Q21:

Discuss the existence of l i m π‘₯ β†’ 2 𝑓 ( π‘₯ ) given

  • AThe limit exists and equals βˆ’ 2 2 .
  • BThe limit exists and equals 3 3 .
  • CThe limit does not exist because l i m π‘₯ β†’ 2 βˆ’ 𝑓 ( π‘₯ ) does not exist.
  • DThe limit does not exist because l i m l i m π‘₯ β†’ 2 π‘₯ β†’ 2 βˆ’ + 𝑓 ( π‘₯ ) β‰  𝑓 ( π‘₯ ) .
  • EThe limit does not exist because l i m π‘₯ β†’ 2 + 𝑓 ( π‘₯ ) does not exist.

Q22:

Suppose that What can be said about the existence of l i m π‘₯ β†’ πœ‹ 𝑓 ( π‘₯ ) ?

  • AThe limit exists and equals βˆ’ 3 .
  • BThe limit does not exist because l i m l i m π‘₯ β†’ πœ‹ π‘₯ β†’ πœ‹ βˆ’ + 𝑓 ( π‘₯ ) β‰  𝑓 ( π‘₯ ) .
  • CThe limit exists and equals 7 2 .

Q23:

Find l i m π‘₯ β†’ βˆ’ 1 𝑓 ( π‘₯ ) given

  • A10
  • B28
  • C βˆ’ 2

Q24:

Discuss the existence of l i m π‘₯ β†’ 0 𝑓 ( π‘₯ ) given

  • AThe limit exists and equals 0.
  • BThe limit does not exist because l i m l i m π‘₯ β†’ 0 π‘₯ β†’ 0 βˆ’ + 𝑓 ( π‘₯ ) β‰  𝑓 ( π‘₯ ) .
  • CThe limit exists and equals 3.
  • DThe limit exists and equals 5.
  • EThe limit exists and equals 7.

Q25:

Given that find l i m π‘₯ β†’ 0 𝑓 ( π‘₯ ) .

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