Worksheet: One-Sided Limits

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In this worksheet, we will practice evaluating one-sided limits graphically and algebraically.

Q1:

Determine l i m 𝑥 → − 4 − 𝑓 ( 𝑥 ) .

Q2:

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

Q3:

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

  • A9
  • B − ∞
  • C0
  • D ∞

Q4:

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

Q5:

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

Q6:

Find l i m → 𝑓 ( 𝑥 ) given 𝑓 ( 𝑥 ) = ⎧ ⎨ ⎩ 5 𝑥 5 𝑥 + 2 5 𝑥 𝑥 0 < 𝑥 < 𝜋 2 , 4 2 9 𝑥 + 𝜋 𝜋 2 < 𝑥 < 𝜋 . c o s s i n i f c o s i f

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

Q7:

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.

Q8:

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

Q9:

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

  • A0
  • B ∞
  • C5
  • D − ∞

Q10:

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

Q11:

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

Q12:

Determine l i m 𝑥 → − 6 − 𝑓 ( 𝑥 ) , if it exists.

  • A2
  • B − 6
  • C0
  • DThe limit does not exist.

Q13:

Use the graph below to find l i m 𝑥 → 8 + 𝑓 ( 𝑥 ) .

  • A3
  • B8
  • C4
  • DThe limit does not exist.

Q14:

Use the graph below to find l i m 𝑥 → 5 + 𝑓 ( 𝑥 ) .

Q15:

Determine l i m 𝑥 → 0 + 𝑓 ( 𝑥 ) , if it exists.

Q16:

Determine l i m 𝑥 → − 4 − 𝑓 ( 𝑥 ) , if it exists.

Q17:

Determine l i m l i m l i m 𝑥 → − 5 𝑥 → − 7 𝑥 → 0 𝑓 ( 𝑥 ) + 𝑓 ( 𝑥 ) + 𝑓 ( 𝑥 ) + − .

Q18:

Determine 𝑓 ( 1 ) − .

Q19:

Determine 𝑓 ( 0 ) − .

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