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Worksheet: Continuity of a Function

Q1:

Find the value of π‘˜ which makes the function 𝑓 continuous at π‘₯ = 3 , given

  • A βˆ’ 2 2 7
  • B βˆ’ 1 2 7
  • C 1 2 7
  • D βˆ’ 1 5 4
  • E βˆ’ 5 4

Q2:

Find the set on which 𝑓 ( π‘₯ ) = √ π‘₯ + 1 4 + √ 3 βˆ’ π‘₯ is continuous.

  • A 𝑓 is continuous on ℝ .
  • B 𝑓 is continuous on ℝ βˆ’ { 1 4 , 3 } .
  • C 𝑓 is continuous on ℝ βˆ’ ( 1 4 , 3 ) .
  • D 𝑓 is continuous on [ βˆ’ 1 4 , 3 ] .
  • E 𝑓 is continuous on ( 1 4 , 3 ) .

Q3:

Which of the following holds for 𝑓 ( π‘₯ ) = ( βˆ’ 7 π‘₯ + 3 ) + 3 5 π‘₯ 7 s i n ?

  • A The function 𝑓 is continuous on ℝ because 3 5 π‘₯ s i n is continuous on ℝ .
  • B The function 𝑓 is continuous on ℝ because ( βˆ’ 7 π‘₯ + 3 ) 7 is continuous on ℝ .
  • C The function 𝑓 is continuous on ℝ βˆ’  π‘₯ ∢ π‘₯ = πœ‹ 2 + 𝑛 πœ‹ , 𝑛 ∈ β„€  .
  • D The function 𝑓 is continuous on ℝ because both of ( βˆ’ 7 π‘₯ + 3 ) 7 and 3 5 π‘₯ s i n are continuous on ℝ .
  • E The function 𝑓 is continuous on ℝ βˆ’ { π‘₯ ∢ π‘₯ = πœ‹ 𝑛 , 𝑛 ∈ β„€ } .

Q4:

Given 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 0 0 π‘₯ βˆ’ 1 0 2 , if possible or necessary, define 𝑓 ( 1 0 ) so that 𝑓 is continuous at π‘₯ = 1 0 .

  • ANo value of 𝑓 ( 1 0 ) will make 𝑓 continuous because l i m π‘₯ β†’ 1 0 𝑓 ( π‘₯ ) does not exist.
  • BThe function is already continuous at π‘₯ = 1 0 .
  • CThe function cannot be made continuous at π‘₯ = 1 0 because 𝑓 ( 1 0 ) is undefined.
  • D 𝑓 ( 1 0 ) = 2 0 makes 𝑓 continuous at π‘₯ = 1 0 .

Q5:

Suppose What can be said of the continuity of 𝑓 at π‘₯ = 0 ?

  • A The function is discontinuous at π‘₯ = 0 because l i m π‘₯ β†’ 0 𝑓 ( π‘₯ ) does not exist.
  • B The function is continuous on ℝ .
  • C The function is discontinuous at π‘₯ = 0 because 𝑓 ( 0 ) is undefined.
  • D The function 𝑓 is continuous at π‘₯ = 0 .
  • E The function is discontinuous at π‘₯ = 0 because 𝑓 ( 0 ) β‰  𝑓 ( π‘₯ ) l i m π‘₯ β†’ 0 .

Q6:

Determine whether the function represented by the graph is continuous or discontinuous.

  • Acontinuous
  • Bdiscontinuous

Q7:

Find the set on which 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 2 2 π‘₯ βˆ’ 2 π‘₯ βˆ’ 6 3 2 is continuous.

  • A 𝑓 ( π‘₯ ) is continuous on ℝ .
  • B 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’ { 2 2 } .
  • C 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’ { βˆ’ 9 , 7 } .
  • D 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’ { 9 , βˆ’ 7 } .

Q8:

Find the set on which 𝑓 ( π‘₯ ) = π‘₯ + 3 π‘₯ βˆ’ 3 π‘₯ + 2 2 is continuous.

  • A 𝑓 ( π‘₯ ) is continuous on ℝ .
  • B 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’ { βˆ’ 3 } .
  • C 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’ { βˆ’ 2 , βˆ’ 1 } .
  • D 𝑓 ( π‘₯ ) is continuous on ℝ βˆ’ { 2 , 1 } .

Q9:

Given 𝑓 ( π‘₯ ) = βˆ’ 9 π‘₯ + π‘₯ βˆ’ 2 2 , what can be said of the continuity of 𝑓 at π‘₯ = βˆ’ 7 ?

  • AThe function is discontinuous at π‘₯ = βˆ’ 7 because l i m π‘₯ β†’ βˆ’ 7 𝑓 ( π‘₯ ) does not exist.
  • BThe function is continuous on ℝ .
  • CThe function is discontinuous at π‘₯ = βˆ’ 7 because 𝑓 ( βˆ’ 7 ) is undefined.
  • DThe function is continuous at π‘₯ = βˆ’ 7 .
  • EThe function is discontinuous at π‘₯ = βˆ’ 7 because 𝑓 ( βˆ’ 7 ) β‰  𝑓 ( π‘₯ ) . l i m π‘₯ β†’ βˆ’ 7

Q10:

Use continuity to evaluate l i m l n π‘₯ β†’ 5 2 ο€Ύ 5 + 2 π‘₯ 6 βˆ’ π‘₯  .

  • A l n 5 4
  • B l n 5 6
  • C55
  • D l n 5 5

Q11:

Use continuity to evaluate l i m l n π‘₯ β†’ 5 2 ο€Ύ 3 + π‘₯ 3 + 5 π‘₯  .

  • A l n 2 8
  • B l n 5 6
  • C1
  • D0

Q12:

Determine the value of π‘Ž that makes the function 𝑓 continuous at π‘₯ = 9 , given

  • A 3 4
  • B 1 1 6
  • C1
  • D 1 8

Q13:

Suppose Discuss whether it is possible to define 𝑓 ( 6 ) to obtain a function that is continuous at this point.

  • AThe function is continuous at π‘₯ = 6 .
  • BThe function 𝑓 can be defined to be continuous at π‘₯ = 6 as
  • CThe function 𝑓 cannot be defined to be continuous at π‘₯ = 6 because l i m π‘₯ β†’ 6 𝑓 ( π‘₯ ) does not exist.
  • DThe function 𝑓 can be defined to be continuous at π‘₯ = 6 as
  • EThe function 𝑓 cannot be defined to be continuous at π‘₯ = 6 because 𝑓 ( 6 ) is undefined.

Q14:

Given that 𝑓 and 𝑔 are continuous functions such that 𝑔 ( 6 ) = 6 and l i m π‘₯ β†’ 6 [ 𝑓 ( π‘₯ ) βˆ’ 9 𝑓 ( π‘₯ ) 𝑔 ( π‘₯ ) ] = 5 , determine 𝑓 ( 6 ) .

  • A βˆ’ 5 5 4
  • B βˆ’ 5 8
  • C5
  • D βˆ’ 5 5 3
  • E 5 7

Q15:

Given that 𝑓 and 𝑔 are continuous functions such that 𝑔 ( 1 ) = βˆ’ 9 and l i m π‘₯ β†’ 1 [ 8 𝑓 ( π‘₯ ) + 9 𝑓 ( π‘₯ ) 𝑔 ( π‘₯ ) ] = 7 , determine 𝑓 ( 1 ) .

  • A βˆ’ 7 8 1
  • B 7 1 7
  • C 7 8
  • D βˆ’ 7 7 3
  • E βˆ’ 7

Q16:

Given that 𝑓 and 𝑔 are continuous functions such that 𝑔 ( 3 ) = 6 and l i m π‘₯ β†’ 3 [ 2 𝑓 ( π‘₯ ) βˆ’ 𝑓 ( π‘₯ ) 𝑔 ( π‘₯ ) ] = βˆ’ 5 , determine 𝑓 ( 3 ) .

  • A 5 6
  • B βˆ’ 5
  • C βˆ’ 5 2
  • D 5 4
  • E βˆ’ 5 8

Q17:

Use continuity to evaluate l i m π‘₯ β†’ 1 √ 3 π‘₯ + 2 π‘₯ βˆ’ 1 5 2 .

Q18:

Use continuity to evaluate l i m π‘₯ β†’ 6 √ π‘₯ βˆ’ 5 π‘₯ βˆ’ 2 5 2 .

Q19:

Find the set on which 𝑓 ∢ 𝑓 ( π‘₯ ) = √ 5 π‘₯ βˆ’ 1 3 is continuous.

  • A π‘₯ ∈ β„•
  • B π‘₯ ∈ ℝ βˆ’  1 5 
  • C π‘₯ ∈ β„€
  • D π‘₯ ∈ ℝ

Q20:

Find the set on which 𝑓 ∢ 𝑓 ( π‘₯ ) = √ 4 π‘₯ βˆ’ 1 9 3 is continuous.

  • A π‘₯ ∈ β„•
  • B π‘₯ ∈ ℝ βˆ’  1 9 4 
  • C π‘₯ ∈ β„€
  • D π‘₯ ∈ ℝ

Q21:

Find the set on which 𝑓 ( π‘₯ ) = π‘₯ + 9 π‘₯ + 1 c o s is continuous.

  • AThe function 𝑓 is continuous on ℝ βˆ’  π‘₯ ∢ π‘₯ = 3 πœ‹ 2 + 2 𝑛 πœ‹ , 𝑛 ∈ β„€  .
  • BThe function 𝑓 is continuous on ℝ βˆ’ { π‘₯ ∢ π‘₯ = 2 πœ‹ 𝑛 , 𝑛 ∈ β„€ } .
  • CThe function 𝑓 is continuous on ℝ βˆ’  π‘₯ ∢ π‘₯ = πœ‹ 2 + 𝑛 πœ‹ , 𝑛 ∈ β„€  .
  • DThe function 𝑓 is continuous on ℝ βˆ’ { π‘₯ ∢ π‘₯ = πœ‹ + 2 πœ‹ 𝑛 , 𝑛 ∈ β„€ } .
  • EThe function 𝑓 is continuous on ℝ because π‘₯ + 9 and c o s π‘₯ + 1 are continuous on ℝ .

Q22:

Find the set on which 𝑓 ( π‘₯ ) = 7 π‘₯ βˆ’ 8 7 π‘₯ + 7 c o s is continuous.

  • AThe function 𝑓 is continuous on ℝ βˆ’  π‘₯ ∢ π‘₯ = 3 πœ‹ 2 + 2 𝑛 πœ‹ , 𝑛 ∈ β„€  .
  • BThe function 𝑓 is continuous on ℝ βˆ’ { π‘₯ ∢ π‘₯ = 2 πœ‹ 𝑛 , 𝑛 ∈ β„€ } .
  • CThe function 𝑓 is continuous on ℝ βˆ’  π‘₯ ∢ π‘₯ = πœ‹ 2 + 𝑛 πœ‹ , 𝑛 ∈ β„€  .
  • DThe function 𝑓 is continuous on ℝ βˆ’ { π‘₯ ∢ π‘₯ = πœ‹ + 2 πœ‹ 𝑛 , 𝑛 ∈ β„€ } .
  • EThe function 𝑓 is continuous on ℝ because 7 π‘₯ βˆ’ 8 and 7 π‘₯ + 7 c o s are continuous on ℝ .

Q23:

Use continuity to evaluate l i m s i n s i n π‘₯ β†’ πœ‹ 6 ( 2 π‘₯ βˆ’ 7 6 π‘₯ ) .

  • A2
  • B0
  • C βˆ’ 7
  • D √ 3 2
  • E6

Q24:

Use continuity to evaluate l i m s i n s i n π‘₯ β†’ πœ‹ 9 ( 3 π‘₯ + 9 π‘₯ ) .

  • A3
  • B0
  • C1
  • D √ 3 2
  • E9

Q25:

Use continuity to evaluate .

  • A
  • B
  • C
  • D
  • E8