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Worksheet: One-Sided Limits

Q1:

Discuss the existence of l i m 𝑓 ( 𝑥 ) given

  • AThe limit exists and equals 4 𝜋 .
  • BThe limit does not exist because l i m l i m 𝑓 ( 𝑥 ) 𝑓 ( 𝑥 ) .
  • CThe limit exists and equals 𝜋 2 .
  • DThe limit exists and equals 2 𝜋 .
  • EThe limit exists and equals 2 .

Q2:

Find l i m 𝑥 𝜋 𝑓 ( 𝑥 ) given

  • A 4 𝜋
  • B15
  • C 4 2 + 𝜋
  • D 4 2 + 𝜋

Q3:

Find l i m 𝑥 0 + 𝑓 ( 𝑥 ) given

  • A 2 4 + 𝜋
  • B 2 𝜋
  • C 2 𝜋 + 4
  • D 3 1 6
  • E 3 7 6

Q4:

Given 𝑓 ( 𝑥 ) = 4 𝑥 4 4 | 𝑥 1 1 | , find 𝑓 ( 1 1 ) + 𝑓 ( 1 1 ) 2 + 2 .

Q5:

Discuss the existence of l i m 𝑥 4 𝑓 ( 𝑥 ) given

  • A l i m 𝑥 4 𝑓 ( 𝑥 ) exists and equals 5.
  • B l i m 𝑥 4 𝑓 ( 𝑥 ) exists and equals 1 .
  • C l i m 𝑥 4 𝑓 ( 𝑥 ) does not exist because l i m 𝑥 4 + 𝑓 ( 𝑥 ) is undefined.
  • D l i m 𝑥 4 𝑓 ( 𝑥 ) exists and equals 2.

Q6:

Find l i m 𝑥 𝜋 6 𝑓 ( 𝑥 ) given

  • A 8 3 + 𝜋
  • B 1 6 3 𝜋
  • C 5 3 2 + 2
  • D 2 + 5 3 2
  • E 3

Q7:

Find l i m 𝑥 5 𝑓 ( 𝑥 ) , if it exists.

Q8:

Determine l i m 𝑥 9 𝑓 ( 𝑥 ) and l i m 𝑥 9 + 𝑓 ( 𝑥 ) , given that

  • A l i m 𝑥 9 𝑓 ( 𝑥 ) = 0 , l i m 𝑥 9 + 𝑓 ( 𝑥 ) = 0
  • B l i m 𝑥 9 𝑓 ( 𝑥 ) = 0 , l i m 𝑥 9 + 𝑓 ( 𝑥 ) = 1 9
  • C l i m 𝑥 9 𝑓 ( 𝑥 ) does not exist, l i m 𝑥 9 + 𝑓 ( 𝑥 ) does not exist
  • D l i m 𝑥 9 𝑓 ( 𝑥 ) = 0 , l i m 𝑥 9 + 𝑓 ( 𝑥 ) does not exist

Q9:

Determine l i m 𝑥 7 𝑓 ( 𝑥 ) and l i m 𝑥 7 + 𝑓 ( 𝑥 ) , given that

  • A l i m 𝑥 7 𝑓 ( 𝑥 ) = 2 , l i m 𝑥 7 + 𝑓 ( 𝑥 ) = 2
  • B l i m 𝑥 7 𝑓 ( 𝑥 ) = 2 , l i m 𝑥 7 + 𝑓 ( 𝑥 ) does not exist
  • C l i m 𝑥 7 𝑓 ( 𝑥 ) does not exist, l i m 𝑥 7 + 𝑓 ( 𝑥 ) does not exist
  • D l i m 𝑥 7 𝑓 ( 𝑥 ) = 2 , l i m 𝑥 7 + 𝑓 ( 𝑥 ) = 1 2

Q10:

Find l i m 𝑥 9 2 2 + 𝑥 + 1 8 𝑥 + 8 1 𝑥 7 𝑥 1 8 .

  • A9
  • B
  • C0
  • D

Q11:

Find l i m l n 𝑥 0 + 5 𝑥 8 3 𝑥 .

  • A0
  • B
  • C 8 3
  • D

Q12:

Determine the following infinite limit: l i m l n s i n 𝑥 0 + 5 ( 𝑥 ) .

  • A0
  • B
  • C5
  • D

Q13:

Determine the following infinite limit: l i m c o t 𝑥 3 𝜋 + 5 2 𝑥 .

  • A0
  • B
  • C 3 𝜋
  • D
  • E 5

Q14:

Determine l i m 𝑥 4 𝑓 ( 𝑥 ) .

Q15:

Determine the following infinite limit: l i m s e c 𝑥 5 𝜋 2 + 1 𝑥 𝑥 .

  • A 5 𝜋 2
  • B
  • C 1
  • D
  • E0

Q16:

Discuss the existence of l i m 𝑥 1 4 𝑓 ( 𝑥 ) given

  • AThe limit exists and equals 6.
  • BThe limit exists and equals 8 1 6 .
  • CThe limit exists and equals 4.
  • DThe limit exists and equals 48.
  • EThe limit does not exist because l i m 𝑥 1 4 + 𝑓 ( 𝑥 ) is undefined.

Q17:

Find l i m 𝑥 𝜋 6 + 𝑓 ( 𝑥 ) given

  • A 4 5 + 3 𝜋 5
  • B 3 3 2 + 6
  • C 3 3 2 + 6
  • D 8 5 + 3 𝜋 5
  • E9