Worksheet: Limits at Infinity and Unbounded Limits

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 l i m 6 𝑥 𝑥 6 .

  • A 1
  • B 0
  • C
  • D 6

Q2:

Find l i m 1 6 𝑥 + 8 9 𝑥 + 3 .

  • A 1 6 9
  • B 0
  • C
  • D 4 3

Q3:

Find l i m 𝑥 + 3 8 𝑥 + 9 𝑥 + 1 .

  • A 0
  • B 1 8
  • C 3
  • D

Q4:

Find l i m 3 𝑥 8 𝑥 + 4 7 𝑥 ( 2 𝑥 + 7 ) .

  • A 1 1 8
  • B 4 5 1 9 6
  • C 0
  • D

Q5:

Find l i m 3 𝑥 4 𝑥 + 8 .

  • A 3 4
  • B 0
  • C 3 8
  • D

Q6:

Find l i m 3 𝑥 𝑥 𝑥 + 3 𝑥 + 2 𝑥 8 𝑥 𝑥 5 𝑥 8 .

  • A
  • B3
  • C 3
  • D

Q7:

Find l i m 9 8 𝑥 + 6 𝑥 2 𝑥 .

  • A9
  • B5
  • C
  • D
  • E 2

Q8:

Consider the polynomial 𝑓 ( 𝑥 ) = 5 𝑥 + 9 𝑥 2 𝑥 𝑥 + 1 1 .

Which of the following is equal to l i m 𝑓 ( 𝑥 ) ?

  • A 2 𝑥 l i m
  • B 5 𝑥 l i m
  • C l i m 1 1
  • D 𝑥 l i m

Hence, find l i m 𝑓 ( 𝑥 ) .

  • A11
  • B
  • C5
  • D
  • E 2

Q9:

Find l i m 7 𝑥 + 8 𝑥 + 4 5 𝑥 + 3 𝑥 .

Q10:

Consider the rational function 𝑓 ( 𝑥 ) = 3 𝑥 8 𝑥 9 2 𝑥 2 2 .

Which of the following is equal to l i m 𝑥 𝑓 ( 𝑥 ) ?

  • A 3 8 9 + 2 l i m 𝑥 1 𝑥 2
  • B 3 8 9 2 l i m l i m 𝑥 1 𝑥 𝑥 1 𝑥 2
  • C 3 + 8 9 + 2 l i m l i m 𝑥 1 𝑥 𝑥 1 𝑥 2
  • D 3 8 9 2 l i m 𝑥 1 𝑥
  • E 3 8 9 2 l i m 𝑥 1 𝑥 2

Find l i m 𝑥 𝑓 ( 𝑥 ) .

  • A 5 2
  • B 5 7
  • C 3 2
  • D 5 2
  • E 3 2

Q11:

Find l i m 2 𝑥 + 8 𝑥 𝑥 + 9 𝑥 4 2 𝑥 6 𝑥 + 7 𝑥 + 6 𝑥 + 3 .

  • A 4 3
  • B
  • C 4 3
  • D

Q12:

Determine l i m ( 5 𝑥 + 4 ) ( 2 𝑥 + 4 ) ( 𝑥 + 2 ) ( 5 𝑥 + 1 ) .

  • A
  • B0
  • C 8 5
  • D
  • E 2

Q13:

Determine l i m 5 𝑥 + 3 ( 𝑥 5 ) ( 4 𝑥 3 𝑥 ) , if it exists.

  • A 5 4
  • B0
  • C The limit does not exist.
  • D25
  • E 2 5 4

Q14:

Find l i m 𝑥 3 6 𝑥 + 3 𝑥 + 3 3 𝑥 .

  • A
  • B0
  • C
  • D3

Q15:

Determine l i m 6 4 𝑥 + 3 𝑥 + 4 4 𝑥 + 2 𝑥 .

  • A 1 1 0
  • B
  • C

Q16:

Find l i m 𝑛 𝑎 + 1 𝑛 𝑎 .

  • A 4 𝑎
  • B 𝑎
  • C 5 𝑎
  • D 5 𝑎

Q17:

Find the values of 𝑎 and 𝑏 , given that l i m 𝑓 ( 𝑥 ) = 4 and l i m 𝑓 ( 𝑥 ) = 5 , where 𝑓 ( 𝑥 ) = 𝑎 𝑥 5 4 𝑏 𝑥 𝑥 + 4 .

  • A 𝑎 = 1 6 , 𝑏 = 2
  • B 𝑎 = 1 8 , 𝑏 = 2
  • C 𝑎 = 1 6 , 𝑏 = 1 6
  • D 𝑎 = 4 , 𝑏 = 1 6
  • E 𝑎 = 4 , 𝑏 = 2

Q18:

Determine l i m 2 𝑥 𝑥 9 𝑥 + 2 ( 2 𝑥 + 5 ) .

  • A 1 4
  • B
  • C 1
  • D

Q19:

Determine l i m 8 𝑥 + 7 6 | 𝑥 | 2 .

  • A 7 2
  • B
  • C0
  • D
  • E 4 3

Q20:

Find l i m 4 𝑥 3 6 𝑥 + 3 + 8 𝑥 7 𝑥 + 3 9 𝑥 + 5 𝑥 + 5 .

  • A
  • B 8 9
  • C
  • D 2 3
  • E 1 4 9

Q21:

If 𝑓 ( 𝑥 ) is a polynomial function of a fifth degrees, and 𝑔 ( 𝑥 ) is a polynomial function of a fourth degrees, find l i m 𝑔 ( 𝑥 ) 4 𝑥 𝑓 ( 𝑥 ) .

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

Q22:

Find l i m 3 𝑥 5 𝑥 3 + 2 6 .

  • A 5
  • B 1 3 5
  • C 2
  • D 3 5

Q23:

Find the values of 𝑎 and 𝑏 , given that l i m 5 𝑥 2 𝑥 + 3 ( 𝑎 + 4 ) 𝑥 + ( 1 𝑏 ) 𝑥 + 5 𝑥 = .

  • A 𝑎 = 4 , 𝑏 = 1
  • B 𝑎 = 4 , 𝑏 = 1
  • C 𝑎 = 4 , 𝑏 = 1
  • D 𝑎 = 4 , 𝑏 = 1

Q24:

By rationalizing and using limit laws, determine l i m 𝑛 + 3 𝑛 𝑛 .

  • A2
  • B6
  • C 3 2
  • D 1 2
  • E3

Q25:

Find l i m 4 𝑥 + 5 𝑥 + 8 .

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