Worksheet: Limits at Infinity and Unbounded Limits

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In this worksheet, we will practice evaluating limits of a function when x tends to infinity and exploring unbounded limits that tend to infinity as x approaches a certain value.

Q1:

Find lim→∞6𝑥𝑥−6.

  • A−1
  • B∞
  • C6
  • D0

Q2:

Find lim→∞16𝑥+89𝑥+3.

  • A∞
  • B43
  • C169
  • D0

Q3:

Find lim→∞𝑥+38𝑥+9𝑥+1.

  • A∞
  • B18
  • C3
  • D0

Q4:

Find lim→∞3𝑥8𝑥+4−7𝑥(2𝑥+7).

  • A−118
  • B∞
  • C45196
  • D0

Q5:

Find lim→∞−3𝑥−4𝑥+8.

  • A0
  • B−38
  • C∞
  • D34

Q6:

Find lim→∞3𝑥−𝑥−𝑥+3𝑥+2𝑥−8𝑥−𝑥−5𝑥−8.

  • A∞
  • B−∞
  • C−3
  • D3

Q7:

Find lim→∞9−8𝑥+6𝑥−2𝑥.

  • A5
  • B9
  • C∞
  • D−∞
  • E−2

Q8:

Consider the polynomial 𝑓(𝑥)=5𝑥+9𝑥−2𝑥−𝑥+11.

Which of the following is equal to lim→∞𝑓(𝑥)?

  • A−𝑥lim→∞
  • Blim→∞11
  • C−2𝑥lim→∞
  • D5𝑥lim→∞

Hence, find lim→∞𝑓(𝑥).

  • A−2
  • B−∞
  • C5
  • D11
  • E∞

Q9:

Find lim→∞7𝑥+8𝑥+45𝑥+3𝑥.

Q10:

Consider the rational function 𝑓(𝑥)=3𝑥−8𝑥9−2𝑥.

Which of the following is equal to lim→∞𝑓(𝑥)?

  • A3−891𝑥+2lim→∞
  • B3−81𝑥91𝑥−2limlim→∞→∞
  • C3+81𝑥91𝑥+2limlim→∞→∞
  • D3−81𝑥9−2lim→∞
  • E3−891𝑥−2lim→∞

Find lim→∞𝑓(𝑥).

  • A−57
  • B−52
  • C52
  • D32
  • E−32

Q11:

Find lim→∞−2𝑥+8𝑥−𝑥+9𝑥−42𝑥−6𝑥+7𝑥+6𝑥+3.

  • A∞
  • B−43
  • C43
  • D−∞

Q12:

Determine lim→∞(−5𝑥+4)(2𝑥+4)(𝑥+2)(5𝑥+1).

  • A0
  • B−85
  • C−∞
  • D−2
  • E∞

Q13:

Determine lim→∞5𝑥+3(𝑥−5)(4𝑥−3𝑥), if it exists.

  • A254
  • B54
  • C0
  • DThe limit does not exist.
  • E25

Q14:

Find lim→∞𝑥√36𝑥+3𝑥+3−3𝑥.

  • A0
  • B3
  • C∞
  • D−∞

Q15:

Determine lim→∞√64𝑥+3𝑥+4−√4𝑥+2𝑥.

  • A−∞
  • B110
  • C∞

Q16:

Find lim→∞𝑛𝑎+1𝑛−𝑎.

  • A𝑎
  • B5𝑎
  • C5𝑎
  • D4𝑎

Q17:

Find the values of 𝑎 and 𝑏, given that lim→∞𝑓(𝑥)=−4 and lim→𝑓(𝑥)=5, where 𝑓(𝑥)=−𝑎𝑥−54𝑏𝑥−𝑥+4.

  • A𝑎=−18, 𝑏=2
  • B𝑎=−16, 𝑏=−16
  • C𝑎=−16, 𝑏=2
  • D𝑎=−4, 𝑏=2
  • E𝑎=−4, 𝑏=−16

Q18:

Determine lim→∞2𝑥−𝑥−9𝑥+2(−2𝑥+5).

  • A−∞
  • B∞
  • C−14
  • D−1

Q19:

Determine lim→∞8𝑥+76|𝑥|−2.

  • A43
  • B−72
  • C0
  • D∞
  • E−∞

Q20:

Find lim→∞4𝑥−3−6𝑥+3+8𝑥−7𝑥+3−9𝑥+5𝑥+5.

  • A∞
  • B−∞
  • C−89
  • D−149
  • E−23

Q21:

If 𝑓(𝑥) is a polynomial function of a fifth degrees, and 𝑔(𝑥) is a polynomial function of a fourth degrees, find lim→∞𝑔(𝑥)4𝑥𝑓(𝑥).

  • A±∞
  • Breal number≠0
  • Chas no limit
  • Dzero

Q22:

Find lim→∞3𝑥5𝑥−3+2√6.

  • A5
  • B135
  • C2
  • D35

Q23:

Find the values of 𝑎 and 𝑏, given that lim→∞5𝑥−2𝑥+3(𝑎+4)𝑥+(1−𝑏)𝑥+5𝑥=∞.

  • A𝑎=−4, 𝑏=1
  • B𝑎=4, 𝑏=−1
  • C𝑎=4, 𝑏=1
  • D𝑎=−4, 𝑏=−1

Q24:

By rationalizing and using limit laws, determine lim→∞√𝑛+3𝑛−𝑛.

  • A3
  • B2
  • C6
  • D32
  • E12

Q25:

Find lim→∞−4𝑥+5𝑥+8.

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