Worksheet: Limits of a Difference of Powers

In this worksheet, we will practice evaluating limits of a difference of powers.

Q1:

Determine l i m 𝑥 4 .

  • AThe limit does not exist.
  • B 5 1 2 8
  • C 1 3 2 0
  • D 5 2 5 6
  • E 1 1 2 8

Q2:

Find l i m 𝑥 5 𝑥 6 𝑥 + 1 + 𝑥 + 1 𝑥 + 1 .

  • AThe limit does not exist.
  • B 2 2 3
  • C 1 7 3
  • D8
  • E 1 4 3

Q3:

Find l i m 2 𝑥 1 6 + 4 4 𝑥 1 6 𝑥 1 2 𝑥 1 6 4 .

  • A 1 1 6
  • B0
  • C 1 3 2
  • DThe limit does not exist.
  • E4

Q4:

Discuss the existence of l i m 𝑓 ( 𝑥 ) given 𝑓 ( 𝑥 ) = 𝑥 + 5 1 2 𝑥 + 1 2 8 6 < 𝑥 < 0 , 8 𝑥 2 𝑥 8 𝑥 6 𝑥 0 < 𝑥 < 𝜋 2 . i f c o s s i n i f

  • A 9 7
  • B 7 3 6
  • C4
  • DThe limit does not exist.
  • E 3 6 7

Q5:

Find l i m 𝑓 ( 𝑥 ) given 𝑓 ( 𝑥 ) = 𝑥 + 1 𝑥 + 1 4 < 𝑥 < 0 , 7 𝑥 2 𝑥 9 𝑥 2 𝑥 0 < 𝑥 < 𝜋 2 . i f c o s s i n i f

  • A 7 𝜋 1 8 𝜋 1 2
  • B 7 𝜋 1 8 𝜋 2
  • Cdoes not exist
  • D 𝜋 1 8 𝜋 1 2
  • E 7 𝜋 9 𝜋 6

Q6:

Find l i m 𝑓 ( 𝑥 ) given 𝑓 ( 𝑥 ) = 𝑥 + 1 𝑥 + 1 8 < 𝑥 < 0 , 4 𝑥 2 𝑥 9 𝑥 5 𝑥 0 < 𝑥 < 𝜋 2 . i f c o s s i n i f

  • A1
  • B 2 7
  • C 7 5
  • D 4 9
  • E The limit does not exist.

Q7:

Find l i m 𝑥 + 1 𝑥 + 1 𝑥 .

  • Ahas no limit
  • B 1 4 2
  • C 1 3 4 2
  • D 1 3 4 2
  • E 1 4 2

Q8:

Find l i m 𝑥 + 𝑥 2 𝑥 1 .

Q9:

Find l i m 5 𝑥 4 1 9 𝑥 8 1 .

  • A 5 6 3
  • B0
  • C 2 5 7
  • D 2 5 6 3
  • E 5 7

Q10:

Find l i m ( 3 𝑥 4 ) + 2 𝑥 2 0 𝑥 2 .

Q11:

Find l i m 𝑥 1 ( 𝑥 1 ) × 1 𝑥 1 .

  • AThe limit does not exist.
  • B 2 8 1
  • C 1 2 8 8 1
  • D 4 2 7

Q12:

Find l i m 𝑥 8 𝑥 5 1 2 .

  • A 1 6 4
  • B 1 1 9 2
  • C 1 3
  • D0

Q13:

Given that l i m l i m 𝑥 + 8 𝑥 + 2 = 𝑥 𝑘 𝑥 𝑘 , determine all the possible values of 𝑘 .

  • A 𝑘 = 6 , 𝑘 = 6
  • B 𝑘 = 8
  • C 𝑘 = 3 , 𝑘 = 3
  • D 𝑘 = 6
  • E 𝑘 = 1 , 𝑘 = 1

Q14:

Find l i m ( 𝑥 + 1 ) 1 𝑥 + 2 1 6 6 .

  • A0
  • B36
  • C540
  • D108

Q15:

Find l i m 𝑥 + 𝑥 2 𝑥 1 .

  • A 1 2 8
  • B 7 3 3
  • CThe limit doesn’t exist.
  • D 4 3 3
  • E 1 1 3 2

Q16:

Find l i m 3 6 𝑥 1 6 𝑥 1 .

Q17:

Find l i m 𝑥 𝑥 𝑥 𝑥 .

  • A 6 1 7
  • B 8 1 9
  • CThe limit does not exist.
  • D 1 7 6
  • E 7 1 8

Q18:

Find l i m 𝑥 6 2 5 𝑥 1 2 5 .

  • A 1 5 4
  • Bhas no limit
  • C20
  • D5
  • E 2 0 3

Q19:

Find l i m ( 𝑥 4 ) + 8 𝑥 2 .

Q20:

Find l i m 𝑥 2 4 3 𝑥 2 , 1 8 7 .

  • A 5 6 3
  • B 7 4 5
  • C 1 9
  • D 5 6 3
  • EThe limit does not exist.

Q21:

Find l i m 𝑥 4 8 1 𝑥 7 .

Q22:

Find l i m 𝑥 + 1 0 2 𝑥 6 .

  • A16
  • B 1 2
  • CThe limit does not exist.
  • D 1 3 2
  • E 1 3 2

Q23:

If l i m 𝑥 8 1 𝑥 2 7 = 𝑙 , what are the values of 𝑛 and 𝑙 ?

  • A 𝑛 = 3 , 𝑙 = 4 9
  • B 𝑛 = 4 , 𝑙 = 4
  • C 𝑛 = 3 , 𝑙 = 1 3
  • D 𝑛 = 4 , 𝑙 = 1 3

Q24:

Determine the value of l i m 𝑥 2 𝑥 2 to the nearest hundredth.

Q25:

Given that 𝑓 ( 𝑥 ) = 1 𝑥 , find l i m 𝑓 ( 𝑥 ) 𝑓 ( 2 ) 𝑥 4 .

  • A 3 6 4
  • B 6 4 3
  • CThe limit does not exist.
  • D 3 6 4
  • E 6 4 3

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