Worksheet: Existence of Limits

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

Q1:

Discuss the existence of l i m → 𝑓 ( 𝑥 ) given 𝑓 ( 𝑥 ) = − 1 5 𝑥 − 1 5 𝑥 < − 1 5 , 𝑥 − 1 5 𝑥 ≥ − 1 5 . i f i f

  • AThe limit does not exist because l i m → 𝑓 ( 𝑥 ) exists, but l i m → 𝑓 ( 𝑥 ) does not exist.
  • BThe limit does not exist because l i m → 𝑓 ( 𝑥 ) exists, but l i m → 𝑓 ( 𝑥 ) does not exist.
  • CThe limit does not exist because both l i m → 𝑓 ( 𝑥 ) and l i m → 𝑓 ( 𝑥 ) exist, but are not equal.
  • DThe limit exists and equals 210.
  • EThe limit exists and equals − 1 5 .

Q2:

Discuss the existence of l i m → 𝑓 ( 𝑥 ) given 𝑓 ( 𝑥 ) = 𝑥 − 4 𝑥 < − 1 , 2 0 𝑥 > − 1 . i f i f

  • AThe limit exists and equals − 2 0 .
  • BThe limit exists and equals 20.
  • CThe limit does not exist because 𝑓 ( − 1 ) does not exist.
  • DThe limit does not exist because 𝑓 ( − 1 ) ≠ 𝑓 ( − 1 ) .

Q3:

Discuss the existence of l i m → 𝑓 ( 𝑥 ) given 𝑓 ( 𝑥 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 2 6 2 2 − 𝑥 𝑥 < 4 , √ 𝑥 − 2 𝑥 − 4 𝑥 > 4 . i f i f

  • AThe limit exists and equals 1 3 9 .
  • BThe limit exists and equals 5.
  • CThe limit does not exist because l i m → 𝑓 ( 𝑥 ) does not exist.
  • DThe limit does not exist because l i m l i m → → 𝑓 ( 𝑥 ) ≠ 𝑓 ( 𝑥 ) .
  • EThe limit does not exist because l i m → 𝑓 ( 𝑥 ) does not exist.

Q4:

Discuss the existence of l i m → 𝑓 ( 𝑥 ) given 𝑓 ( 𝑥 ) = ⎧ ⎨ ⎩ 1 0 𝑥 1 0 𝑥 − 𝜋 3 < 𝑥 < 0 , 𝑥 0 < 𝑥 < 𝜋 3 . t a n i f c o s i f

  • AThe limit exists and equals 0.
  • BThe limit does not exist because l i m l i m → → 𝑓 ( 𝑥 ) ≠ 𝑓 ( 𝑥 ) .
  • CThe limit exists and equals 𝜋 3 .
  • DThe limit exists and equals 1.
  • EThe limit exists and equals − 𝜋 3 .

Q5:

Discuss the existence of l i m → 𝑓 ( 𝑥 ) given 𝑓 ( 𝑥 ) = 3 𝑥 + 7 𝑥 𝑥 + 𝑥 𝑥 < 0 , 5 𝑥 𝑥 > 0 . t a n s i n i f c o s i f

  • AThe limit exists and equals 0.
  • BThe limit does not exist because l i m l i m → → 𝑓 ( 𝑥 ) ≠ 𝑓 ( 𝑥 ) .
  • CThe limit exists and equals 3.
  • DThe limit exists and equals 5.
  • EThe limit exists and equals 7.

Q6:

Determine l i m → 𝑓 ( 𝑥 ) if it exists, where 𝑓 ( 𝑥 ) = ⎧ ⎨ ⎩ 2 𝑥 + 8 𝑥 2 𝑥 − 𝜋 2 < 𝑥 < 0 , 5 𝑥 𝑥 0 < 𝑥 < 𝜋 2 . s i n t a n i f t a n i f

  • A 1 5
  • B1
  • C4
  • D5

Q7:

Discuss the existence of the limit as 𝑥 → 𝜋 given 𝑓 ( 𝑥 ) = − 9 𝑥 𝑥 − 𝜋 𝑥 < 𝜋 , − 9 𝑥 𝑥 > 𝜋 . s i n i f c o s i f

  • AThe limit exists and equals − 9 .
  • BThe limit does not exist because l i m l i m → → 𝑓 ( 𝑥 ) ≠ 𝑓 ( 𝑥 ) .
  • CThe limit exists and equals − 1 8 .
  • DThe limit exists and equals 9 .

Q8:

Determine l i m → 𝑓 ( 𝑥 ) , where 𝑓 ( 𝑥 ) = ⎧ ⎨ ⎩ ( 6 𝑥 − 5 4 ) 𝑥 − 9 𝑥 < 9 , 6 𝜋 𝑥 2 𝑥 > 9 . s i n i f s i n i f

Q9:

Given that 𝑓 ( 𝑥 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 4 𝑥 1 − 𝑥 𝑥 < 1 , 2 𝜋 𝑥 1 − 𝑥 𝑥 > 1 . c o t i f s i n i f determine l i m → 𝑓 ( 𝑥 ) .

  • A 𝜋 2
  • B2
  • C 1 2
  • D 2 𝜋

Q10:

Describe the limit as 𝑥 → 0 for the following function: 𝑓 ( 𝑥 ) = ⎧ ⎪ ⎨ ⎪ ⎩ √ 1 − 7 𝑥 5 𝑥 𝑥 < 0 , √ 8 𝑥 − 1 7 𝑥 𝑥 > 0 . c o s i f s e c i f

  • AThe limit exists and equals − 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 → → 𝑓 ( 𝑥 ) ≠ 𝑓 ( 𝑥 ) .
  • EThe limit exists and equals − 8 7 .

Q11:

Suppose that 𝑓 ( 𝑥 ) = ⎧ ⎪ ⎨ ⎪ ⎩ ( 𝑥 + 1 ) 𝑥 + 7 𝑥 + 6 𝑥 < − 1 , √ 𝑥 + 2 − 1 √ 5 𝑥 + 6 − 1 𝑥 > − 1 . t a n i f i f What can be said about the existence of l i m → 𝑓 ( 𝑥 ) ?

  • A The limit exists and equals 1.
  • B The limit does not exist because l i m l i m → → 𝑓 ( 𝑥 ) ≠ 𝑓 ( 𝑥 ) .
  • C The limit exists and equals 0.
  • D The limit exists and equals 1 5 .

Q12:

Given that 𝑓 ( 𝑥 ) = ⎧ ⎪ ⎨ ⎪ ⎩ √ 𝑥 9 𝑥 3 𝑥 𝑥 < 0 , ( 2 𝑥 ) 2 𝑥 𝑥 > 0 , t a n i f s i n s i n i f find l i m → 𝑓 ( 𝑥 ) .

  • A 2 3
  • B5
  • C 3 2
  • D1

Q13:

Suppose that 𝑓 ( 𝑥 ) = ⎧ ⎪ ⎨ ⎪ ⎩ − 2 0 𝑥 1 − ( 𝑥 + 𝑥 ) 𝑥 < 0 , ( 𝑥 + 2 ) − 3 2 𝑥 + 8 𝑥 𝑥 > 0 . s i n c o s i f i f What can be said about the existence of l i m → 𝑓 ( 𝑥 ) .

  • A The limit exists and equals 2.
  • B The limit does not exist because l i m l i m → → 𝑓 ( 𝑥 ) ≠ 𝑓 ( 𝑥 ) .
  • C The limit exists and equals − 2 0 .
  • D The limit exists and equals 10.

Q14:

Discuss the existence of l i m → 𝑓 ( 𝑥 ) given 𝑓 ( 𝑥 ) = 𝑥 | 𝑥 + 7 | − 3 .

  • A The limit does not exist because l i m l i m → → 𝑓 ( 𝑥 ) ≠ 𝑓 ( 𝑥 ) .
  • B The limit exists and equals − 3 .
  • C The limit does not exist because l i m → 𝑓 ( 𝑥 ) does not exist.
  • D The limit exists and equals − 9 .
  • E The limit does not exist because l i m → 𝑓 ( 𝑥 ) does not exist.

Q15:

Given that 𝑓 ( 𝑥 ) = 4 + 𝑥 + 3 𝑥 | 𝑥 + 3 | − 3 < 𝑥 < 0 , 2 𝑥 + 4 0 < 𝑥 < 2 , i f i f find l i m → 𝑓 ( 𝑥 ) .

Q16:

Discuss the existence of l i m → 𝑓 ( 𝑥 ) given 𝑓 ( 𝑥 ) = 1 6 𝑥 + | 𝑥 − 2 | 2 − 𝑥 𝑥 < 2 , 𝑥 + 5 𝑥 > 2 . i f i f

  • AThe limit exists and equals − 2 2 .
  • BThe limit exists and equals 3 3 .
  • CThe limit does not exist because l i m → 𝑓 ( 𝑥 ) does not exist.
  • DThe limit does not exist because l i m l i m → → 𝑓 ( 𝑥 ) ≠ 𝑓 ( 𝑥 ) .
  • EThe limit does not exist because l i m → 𝑓 ( 𝑥 ) does not exist.

Q17:

Find l i m → 𝑓 ( 𝑥 ) given 𝑓 ( 𝑥 ) = 5 𝑥 + 3 𝑥 < − 1 , 2 𝑥 − 1 < 𝑥 < 5 , 𝑥 + 4 𝑥 > 5 . i f i f i f

  • A10
  • B28
  • C − 2

Q18:

Discuss the existence of l i m → 𝑓 ( 𝑥 ) given 𝑓 ( 𝑥 ) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 3 𝑥 − 𝜋 2 < 𝑥 < 0 , 3 𝑥 + 1 0 < 𝑥 < 3 , 𝑥 − 2 7 𝑥 − 3 𝑥 > 3 . c o s i f i f i f

  • A l i m → 𝑓 ( 𝑥 ) does not exist because l i m → 𝑓 ( 𝑥 ) exist, but l i m → 𝑓 ( 𝑥 ) does not exist.
  • B l i m → 𝑓 ( 𝑥 ) does not exist because l i m l i m → → 𝑓 ( 𝑥 ) ≠ 𝑓 ( 𝑥 ) .
  • C l i m → 𝑓 ( 𝑥 ) does not exist because l i m → 𝑓 ( 𝑥 ) exist, but l i m → 𝑓 ( 𝑥 ) does not exist.
  • D l i m → 𝑓 ( 𝑥 ) exist and equals 3.

Q19:

Suppose that 𝑓 ( 𝑥 ) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ − 3 9 𝑥 − 𝜋 6 < 𝑥 < 0 , 7 𝑥 − 4 8 𝑥 2 𝑥 0 < 𝑥 < 𝜋 , 7 2 𝑥 > 𝜋 . c o s i f s i n i f i f What can be said about the existence of l i m → 𝑓 ( 𝑥 ) ?

  • AThe limit exists and equals − 3 .
  • BThe limit does not exist because l i m l i m → → 𝑓 ( 𝑥 ) ≠ 𝑓 ( 𝑥 ) .
  • CThe limit exists and equals 7 2 .

Q20:

Given that 𝑓 ( 𝑥 ) = 2 5 𝑥 4 𝑥 𝑥 < 0 , 𝑥 + 𝑎 𝑥 > 0 s i n c o t i f i f and l i m → 𝑓 ( 𝑥 ) exists, determine the possible values of 𝑎 .

  • A 2 5 4 , − 2 5 4
  • B 2 5 , − 2 5
  • C 4 2 5 , − 4 2 5
  • D 5 2 , − 5 2

Q21:

Given that the function 𝑓 ( 𝑥 ) = 𝑥 + 𝑎 𝑥 + 𝑏 𝑥 − 5 𝑥 + 6 𝑥 < 2 , 6 𝑥 𝑥 > 2 i f i f has a limit when 𝑥 = 2 , determine the values of 𝑎 and 𝑏 .

  • A 𝑎 = − 1 2 , 𝑏 = 2 8
  • B 𝑎 = − 8 , 𝑏 = − 1 2
  • C 𝑎 = − 1 6 , 𝑏 = 3 6
  • D 𝑎 = − 1 6 , 𝑏 = 2 8

Q22:

Given that 𝑓 ( 𝑥 ) = ⎧ ⎨ ⎩ − 1 4 𝑥 𝜋 − 2 𝑥 𝑥 < 𝜋 2 , − 4 + ( 𝜋 − 𝑥 ) 𝑥 > 𝜋 2 , c o t i f s i n i f find l i m → 𝑓 ( 𝑥 ) , if it exists.

  • A − 3
  • B1
  • C − 4
  • Ddoes not exist
  • E − 7

Q23:

Discuss the existence of l i m → 𝑓 ( 𝑥 ) given 𝑓 ( 𝑥 ) = ⎧ ⎨ ⎩ ( 𝑥 + 9 ) 𝑥 + 9 𝑥 < − 9 , 𝜋 4 ( 𝑥 + 1 0 ) 𝑥 > − 9 . s i n i f t a n i f

  • AThe limit exists and equals − 1 0 .
  • BThe limit exists and equals − 9 .
  • CThe limit does not exist because l i m l i m → → 𝑓 ( 𝑥 ) ≠ 𝑓 ( 𝑥 ) .
  • DThe limit exists and equals 1.

Q24:

Discuss the existence of l i m → 𝑓 ( 𝑥 ) given 𝑓 ( 𝑥 ) = 5 𝑥 7 𝑥 𝑥 ≤ 0 , 4 𝑥 + 2 1 0 𝑥 𝑥 > 0 . c o t i f s i n c o s i f

  • 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 → → 𝑓 ( 𝑥 ) ≠ 𝑓 ( 𝑥 ) .

Q25:

Discuss the existence of l i m → 𝑓 ( 𝑥 ) given 𝑓 ( 𝑥 ) = ⎧ ⎨ ⎩ 1 0 𝑥 + 8 𝑥 2 𝑥 − 𝜋 2 < 𝑥 < 0 , 9 𝑥 𝑥 0 < 𝑥 < 𝜋 2 . s i n t a n i f t a n i f

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

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