Worksheet: Limits by Direct Substitution

In this worksheet, we will practice using the direct substitution method to evaluate limits.

Q1:

Determine l i m 𝑥 5 2 ( 9 𝑥 6 𝑥 9 ) .

Q2:

Determine l i m 𝑥 9 2 4 𝑥 9 𝑥 + 1 .

  • A 1 1 2
  • B244
  • C 9 3
  • D 2 6 1

Q3:

Find l i m 𝑥 1 ( 3 0 ) .

Q4:

Determine l i m s i n 𝑥 𝜋 4 9 𝑥 5 𝑥 .

  • A 1 8 2 𝜋
  • B 3 6 5 𝜋
  • C 9 2 1 0 𝜋
  • D 1 8 2 5 𝜋
  • E 2 2 5 𝜋

Q5:

Given 𝑓 ( 𝑥 ) = | 𝑥 + 1 1 | | 𝑥 1 8 | , find l i m 𝑥 4 𝑓 ( 𝑥 ) .

Q6:

Find an exact expression for l i m 𝑥 𝑥 + 2 𝑥 + 7 𝑥 + 1 using the limit laws.

  • A 6 7 5 5
  • B 8 3 5 5
  • C 5 9 5 5
  • D 1 5 5
  • E 5 5

Q7:

Find l i m 𝑥 8 7 𝑥 + 7 3 𝑥 3 .

  • A 8
  • B 7 3
  • C 2 1 8
  • D 3

Q8:

Find .

  • A8
  • B4
  • C
  • D0

Q9:

Find l i m c o s 𝑥 1 2 2 𝑥 ( 2 4 𝑥 ) 𝑥 + 𝑥 .

  • A 0
  • B 1 3
  • C 1 2
  • D 2 3

Q10:

Find l i m t a n s i n c o s 𝑥 0 7 3 ( 3 𝑥 ) 7 ( 5 𝑥 ) + 5 ( 6 𝑥 ) .

  • A 9 6 5
  • B0
  • C7
  • D 7 5
  • E 3 1 1

Q11:

Find l i m 𝑥 1 1 2 2 𝑥 + 1 0 𝑥 1 1 𝑥 + 1 1 𝑥 .

  • A 1 0 1 1
  • B 5 5 1 1
  • C11
  • D 1 1 0 1 1
  • E 1 1

Q12:

Given that l i m 𝑥 6 6 𝑓 ( 𝑥 ) + 4 𝑥 + 1 = 4 , find l i m 𝑥 6 𝑓 ( 𝑥 ) .

Q13:

Find l i m 𝑥 5 7 1 ( 𝑥 5 ) .

  • A 5
  • B 1
  • C0
  • DThe limit does not exist.

Q14:

Determine the following infinite limit: l i m 𝑥 8 8 9 𝑥 ( 𝑥 8 ) .

  • A9
  • B
  • C8
  • D
  • E0

Q15:

Find l i m 𝑥 4 𝑥 3 𝑥 + 2 𝑥 4 .

  • A 1 4
  • B4
  • C 1 2
  • D 1 4
  • EThe limit does not exist.

Q16:

If the function 𝑓 ( 𝑥 ) = | 𝑥 + 9 | | 𝑥 1 2 | , find l i m 𝑥 1 2 𝑓 ( 𝑥 ) .

Q17:

If the function 𝑓 ( 𝑥 ) = | 𝑥 + 1 | | 𝑥 5 | , find l i m 𝑥 1 𝑓 ( 𝑥 ) .

Q18:

Find l i m 𝑥 4 3 2 𝑥 𝑥 + 2 𝑥 + 7 𝑥 + 1 to 4 decimal places by considering 𝑓 ( 𝑥 ) 𝑛 at 𝑥 = 4 . 1 , 𝑥 = 4 . 0 1 , 𝑥 = 4 . 0 0 1 , 1 2 3 . What is the first 𝑛 that you can use?

  • A 2 8 . 1 7 4 4 , 𝑛 = 5
  • B 2 8 . 1 7 4 4 , 𝑛 = 6
  • C 2 8 . 1 7 4 4 , 𝑛 = 4
  • D 2 8 . 1 7 4 4 , 𝑛 = 7
  • E 2 8 . 1 7 4 4 , 𝑛 = 3

Q19:

Determine l i m 𝑥 4 2 2 𝑥 9 𝑥 + 2 8 𝑥 3 𝑥 9 .

  • AThe limit does not exist.
  • B 5 4 1 2 5
  • C50
  • D 5 4 1 2 5
  • E 1 4 9

Q20:

Given that l i m 𝑥 3 2 𝑓 ( 𝑥 ) 4 𝑥 = 4 , determine l i m 𝑥 3 𝑓 ( 𝑥 ) 𝑥 .

Q21:

Determine l i m c o s 𝑥 𝜋 3 7 𝑥 𝑥 .

  • A3
  • B 7 𝜋 3
  • C 𝜋 6
  • D 7 𝜋 6

Q22:

Find l i m t a n s i n c o s 𝑥 0 1 5 𝑥 5 𝑥 5 𝑥 .

Q23:

Find l i m c o s 𝑥 0 6 3 𝑥 5 𝑥 .

Q24:

Determine l i m 𝑥 5 1 𝑥 + 7 1 7 𝑥 .

  • A 1 2
  • B 1 7
  • C 5 1 4
  • D 1 1 4

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