Lesson Worksheet: Limits at Infinity Mathematics • Higher Education

In this worksheet, we will practice evaluating limits of a function when x tends to infinity.

Q1:

Find limο—β†’βˆžοŠ¨6π‘₯π‘₯βˆ’6.

  • Aβˆ’1
  • B∞
  • C6
  • D0

Q2:

Find limο—β†’βˆžο„ž16π‘₯+89π‘₯+3.

  • A∞
  • B43
  • C169
  • D0

Q3:

Find limο—β†’βˆžοŠ¨οŠ©π‘₯+38π‘₯+9π‘₯+1.

  • A∞
  • B18
  • C3
  • D0

Q4:

Find limο—β†’βˆžοŠ¨οŠ¨ο€Ώ3π‘₯8π‘₯+4βˆ’7π‘₯(2π‘₯+7).

  • Aβˆ’118
  • B∞
  • C45196
  • D0

Q5:

Find limο—β†’βˆžοŠ¨οŠ¨βˆ’3π‘₯βˆ’4π‘₯+8.

  • A0
  • Bβˆ’38
  • C∞
  • D34

Q6:

Find limο—β†’βˆžοŠͺοŠͺ3π‘₯βˆ’π‘₯βˆ’π‘₯+3π‘₯+2π‘₯βˆ’8π‘₯βˆ’π‘₯βˆ’5π‘₯βˆ’8.

  • A∞
  • Bβˆ’βˆž
  • Cβˆ’3
  • D3

Q7:

Find limο—β†’οŠ±βˆžοŠ¨οŠ©9βˆ’8π‘₯+6π‘₯βˆ’2π‘₯.

  • A5
  • B9
  • C∞
  • Dβˆ’βˆž
  • Eβˆ’2

Q8:

Consider the polynomial 𝑓(π‘₯)=5π‘₯+9π‘₯βˆ’2π‘₯βˆ’π‘₯+11οŠͺ.

Which of the following is equal to limο—β†’βˆžπ‘“(π‘₯)?

  • Aβˆ’π‘₯limο—β†’βˆž
  • Blimο—β†’βˆž11
  • Cβˆ’2π‘₯limο—β†’βˆžοŠ¨
  • D5π‘₯limο—β†’βˆžοŠͺ

Hence, find limο—β†’βˆžπ‘“(π‘₯).

  • Aβˆ’2
  • Bβˆ’βˆž
  • C5
  • D11
  • E∞

Q9:

Find limο—β†’βˆžοŠ¨οŠ©οŠ¨7π‘₯+8π‘₯+45π‘₯+3π‘₯.

Q10:

Consider the rational function 𝑓(π‘₯)=3π‘₯βˆ’8π‘₯9βˆ’2π‘₯.

Which of the following is equal to limο—β†’οŠ±βˆžπ‘“(π‘₯)?

  • A3βˆ’891π‘₯+2limο—β†’οŠ±βˆžοŠ¨
  • B3βˆ’81π‘₯91π‘₯βˆ’2limlimο—β†’οŠ±βˆžο—β†’οŠ±βˆžοŠ¨
  • C3+81π‘₯91π‘₯+2limlimο—β†’οŠ±βˆžο—β†’οŠ±βˆžοŠ¨
  • D3βˆ’81π‘₯9βˆ’2limο—β†’οŠ±βˆž
  • E3βˆ’891π‘₯βˆ’2limο—β†’οŠ±βˆžοŠ¨

Find limο—β†’οŠ±βˆžπ‘“(π‘₯).

  • Aβˆ’57
  • Bβˆ’52
  • C52
  • D32
  • Eβˆ’32

Q11:

Find limο—β†’βˆžοŠ±οŠͺοŠͺοŠ±οŠ©οŠ±οŠ¨οŠ±οŠ§βˆ’2π‘₯+8π‘₯βˆ’π‘₯+9π‘₯βˆ’42π‘₯βˆ’6π‘₯+7π‘₯+6π‘₯+3.

  • A∞
  • Bβˆ’43
  • C43
  • Dβˆ’βˆž

Q12:

Determine limο—β†’βˆžοŠ©οŠ¨οŠ¨(βˆ’5π‘₯+4)(2π‘₯+4)(π‘₯+2)(5π‘₯+1).

  • A0
  • Bβˆ’85
  • Cβˆ’βˆž
  • Dβˆ’2
  • E∞

Q13:

Determine limο—β†’βˆžοŠ¨οŠ¨οŠ¨οŠ¨ο€Ή5π‘₯+3(π‘₯βˆ’5)(4π‘₯βˆ’3π‘₯), if it exists.

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

Q14:

Find limο—β†’βˆžοŠ¨π‘₯ο€»βˆš36π‘₯+3π‘₯+3βˆ’3π‘₯.

  • A0
  • B3
  • C∞
  • Dβˆ’βˆž

Q15:

Determine limο—β†’βˆžοŠ¨οŠ¨ο€»βˆš64π‘₯+3π‘₯+4βˆ’βˆš4π‘₯+2π‘₯.

  • Aβˆ’βˆž
  • B110
  • C∞

Q16:

Find limοŠβ†’βˆžοŠ«οŠ«οŠ«οŠ«π‘›ο–ο€Όπ‘Ž+1π‘›οˆβˆ’π‘Žο’.

  • Aπ‘ŽοŠ«
  • B5π‘ŽοŠͺ
  • C5π‘ŽοŠ«
  • D4π‘ŽοŠ«

Q17:

Find the values of π‘Ž and 𝑏, given that limο—β†’βˆžπ‘“(π‘₯)=βˆ’4 and limο—β†’οŠ«π‘“(π‘₯)=5, where 𝑓(π‘₯)=βˆ’π‘Žπ‘₯βˆ’54𝑏π‘₯βˆ’π‘₯+4.

  • Aπ‘Ž=βˆ’18, 𝑏=2
  • Bπ‘Ž=βˆ’16, 𝑏=βˆ’16
  • Cπ‘Ž=βˆ’16, 𝑏=2
  • Dπ‘Ž=βˆ’4, 𝑏=2
  • Eπ‘Ž=βˆ’4, 𝑏=βˆ’16

Q18:

Determine limο—β†’βˆžοŠ©οŠ¨οŠ©2π‘₯βˆ’π‘₯βˆ’9π‘₯+2(βˆ’2π‘₯+5).

  • Aβˆ’βˆž
  • B∞
  • Cβˆ’14
  • Dβˆ’1

Q19:

Determine limο—β†’βˆž8π‘₯+76|π‘₯|βˆ’2.

  • A43
  • Bβˆ’72
  • C0
  • D∞
  • Eβˆ’βˆž

Q20:

Find limο—β†’βˆžοŠ¨οŠ¨ο€Ύ4π‘₯βˆ’3βˆ’6π‘₯+3+8π‘₯βˆ’7π‘₯+3βˆ’9π‘₯+5π‘₯+5.

  • A∞
  • Bβˆ’βˆž
  • Cβˆ’89
  • Dβˆ’149
  • Eβˆ’23

Q21:

If 𝑓(π‘₯) is a polynomial function of the fifth degree and 𝑔(π‘₯) is a polynomial function of the fourth degree, find limο—β†’βˆžοŠ«π‘”(π‘₯)4π‘₯𝑓(π‘₯).

  • A±∞
  • BA real number β‰ 0
  • CZero
  • DIt has no limit.

Q22:

Find limο—β†’βˆžοŠͺοŠͺο€Ύ3π‘₯5π‘₯βˆ’3+2√6οŠο‘.

  • A5
  • B135
  • C2
  • D35

Q23:

Find the values of π‘Ž and 𝑏, given that limο—β†’βˆžοŠ«οŠ¬οŠ«οŠͺ5π‘₯βˆ’2π‘₯+3(π‘Ž+4)π‘₯+(1βˆ’π‘)π‘₯+5π‘₯=∞.

  • Aπ‘Ž=βˆ’4, 𝑏=1
  • Bπ‘Ž=4, 𝑏=βˆ’1
  • Cπ‘Ž=4, 𝑏=1
  • Dπ‘Ž=βˆ’4, 𝑏=βˆ’1

Q24:

By rationalizing and using limit laws, determine limοŠβ†’βˆžοŠ¨βˆšπ‘›+3π‘›βˆ’π‘›.

  • A3
  • B2
  • C6
  • D32
  • E12

Q25:

Find limο—β†’βˆžοŠ¨ο€Όβˆ’4π‘₯+5π‘₯+8.

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