Worksheet: One-Sided Limits

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

Q1:

Determine lim→𝑓(𝑥).

Q2:

Find lim→𝑓(𝑥), if it exists.

Q3:

Find lim→𝑥+18𝑥+81𝑥−7𝑥−18.

  • A∞
  • B9
  • C0
  • D−∞

Q4:

Determine lim→𝑓(𝑥) and lim→𝑓(𝑥), given that 𝑓(𝑥)=𝑥+9𝑥≤−9,1𝑥+9𝑥>−9.ifif

  • Alim→𝑓(𝑥)does not exist,lim→𝑓(𝑥)does not exist
  • Blim→𝑓(𝑥)=0,lim→𝑓(𝑥)does not exist
  • Clim→𝑓(𝑥)=0,lim→𝑓(𝑥)=−19
  • Dlim→𝑓(𝑥)=0,lim→𝑓(𝑥)=0

Q5:

Find lim→𝑓(𝑥) given 𝑓(𝑥)=⎧⎨⎩7𝑥+3𝑥5𝑥−𝜋2<𝑥<0,−52𝑥+70<𝑥<𝜋2.sinsinifcosif

  • A2
  • B−5√32+7
  • C92
  • D3+7𝜋3

Q6:

Find lim→𝑓(𝑥) given 𝑓(𝑥)=⎧⎨⎩5𝑥5𝑥+25𝑥𝑥0<𝑥<𝜋2,429𝑥+𝜋𝜋2<𝑥<𝜋.cossinifcosif

  • A4𝜋
  • B42+𝜋
  • C15
  • D4−2+𝜋

Q7:

Discuss the existence of lim→𝑓(𝑥) given 𝑓(𝑥)=|𝑥−1|+2−1<𝑥<2,𝑥+2𝑥−8𝑥−2𝑥2<𝑥<4.ifif

  • Alim→𝑓(𝑥) does not exist because lim→𝑓(𝑥) is undefined.
  • Blim→𝑓(𝑥) exists and equals 5.
  • Clim→𝑓(𝑥) exists and equals 2.
  • Dlim→𝑓(𝑥) exists and equals −1.

Q8:

Given 𝑓(𝑥)=4𝑥−44|𝑥−11|, find 𝑓(11)+𝑓(11).

Q9:

Determine the following infinite limit: limlnsin→5(𝑥).

  • A∞
  • B−∞
  • C0
  • D5

Q10:

Determine the following infinite limit: limcot→−52𝑥.

  • A0
  • B−∞
  • C−5
  • D3𝜋
  • E∞

Q11:

Determine lim→𝑓(𝑥) and lim→𝑓(𝑥), given that 𝑓(𝑥)=𝑥+9𝑥≤−7,1𝑥+9𝑥>−7.ifif

  • Alim→𝑓(𝑥)does not exist,lim→𝑓(𝑥)does not exist
  • Blim→𝑓(𝑥)=2,lim→𝑓(𝑥)=12
  • Clim→𝑓(𝑥)=2,lim→𝑓(𝑥)does not exist
  • Dlim→𝑓(𝑥)=2,lim→𝑓(𝑥)=2

Q12:

Determine lim→𝑓(𝑥), if it exists.

  • AThe limit does not exist.
  • B0
  • C−6
  • D2

Q13:

Use the graph below to find lim→𝑓(𝑥).

  • AThe limit does not exist.
  • B3
  • C4
  • D8

Q14:

Use the graph below to find lim→𝑓(𝑥).

Q15:

Determine lim→𝑓(𝑥), if it exists.

Q16:

Determine lim→𝑓(𝑥), if it exists.

Q17:

Determine limlimlim→→→𝑓(𝑥)+𝑓(𝑥)+𝑓(𝑥).

Q18:

Determine 𝑓(1).

Q19:

Determine lim→𝑓(𝑥).

Q20:

Use the graph below to find lim→𝑓(𝑥).

Q21:

Using the graph shown, find lim→𝑓(𝑥).

Q22:

Determine lim→𝑓(𝑥).

Q23:

Find lim→𝑓(𝑥) and lim→𝑓(𝑥), where 𝑓(𝑥)=78𝑥<−9,−9𝑥−7𝑥≥−9.ifif

  • Alimlim→→𝑓(𝑥)=74,𝑓(𝑥)=74
  • Blimlim→→𝑓(𝑥)=78,𝑓(𝑥)=78
  • Climlim→→𝑓(𝑥)=78,𝑓(𝑥)=74
  • Dlimlim→→𝑓(𝑥)=74,𝑓(𝑥)=78

Q24:

Determinelim→𝑓(𝑥).

Q25:

Determine lim→𝑓(𝑥).

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