Worksheet: Continuity of a Function

In this worksheet, we will practice determining the continuity of a function at a point on the number line by evaluating limits at that point.

Q1:

Let Find the value of 𝑎 that makes 𝑓 continuous at 𝑥 = 9 .

  • A 6 2 5
  • B 6 2 9
  • C 6 5
  • D 6 2 9
  • E 2 9 6

Q2:

Find the set on which 𝑓 ( 𝑥 ) = 𝑥 + 1 4 + 3 𝑥 is continuous.

  • A 𝑓 is continuous on .
  • B 𝑓 is continuous on { 1 4 , 3 } .
  • C 𝑓 is continuous on ] 1 4 , 3 [ .
  • D 𝑓 is continuous on [ 1 4 , 3 ] .
  • E 𝑓 is continuous on ] 1 4 , 3 [ .

Q3:

Discuss the continuity of the function 𝑓 ( 𝑥 ) = 𝑥 1 𝑥 1 6 on .

  • AThe function is continuous on [ 4 , 4 ] .
  • BThe function is discontinuous on .
  • CThe function is continuous on .
  • DThe function is continuous on { 4 , 4 } .
  • EThe function is continuous on [ 4 , 4 ] .

Q4:

Find the value of 𝑘 which makes the function 𝑓 continuous at 𝑥 = 3 , given

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

Q5:

Discuss the continuity of the function 𝑓 at 𝑥 = 7 , given that

  • A The function is discontinuous at 𝑥 = 7 because 𝑓 ( 7 ) is undefined.
  • B The function is discontinuous at 𝑥 = 7 because l i m 𝑓 ( 𝑥 ) does not exist.
  • C The function is discontinuous at 𝑥 = 7 because 𝑓 ( 7 ) 𝑓 ( 𝑥 ) l i m .
  • D The function is continuous at 𝑥 = 7 .
  • E The function is continuous at all points in { 7 } .

Q6:

Let What can be said of the continuity of 𝑓 at 𝑥 = 7 ?

  • A The function is discontinuous at 𝑥 = 7 because l i m 𝑥 7 𝑓 ( 𝑥 ) does not exist.
  • B The function is discontinuous at 𝑥 = 7 because 𝑓 ( 7 ) is undefined.
  • C The function is discontinuous at 𝑥 = 7 because 𝑓 ( 7 ) 𝑓 ( 𝑥 ) l i m 𝑥 7 .
  • D The function is continuous at 𝑥 = 7 .

Q7:

Find l i m 𝑥 9 𝑓 ( 𝑥 ) and l i m 𝑥 9 + 𝑓 ( 𝑥 ) , where

  • A l i m l i m 𝑥 9 𝑥 9 + 𝑓 ( 𝑥 ) = 7 8 , 𝑓 ( 𝑥 ) = 7 8
  • B l i m l i m 𝑥 9 𝑥 9 + 𝑓 ( 𝑥 ) = 7 4 , 𝑓 ( 𝑥 ) = 7 8
  • C l i m l i m 𝑥 9 𝑥 9 + 𝑓 ( 𝑥 ) = 7 4 , 𝑓 ( 𝑥 ) = 7 4
  • D l i m l i m 𝑥 9 𝑥 9 + 𝑓 ( 𝑥 ) = 7 8 , 𝑓 ( 𝑥 ) = 7 4

Q8:

Find the values of 𝑐 which make the function 𝑓 continuous at 𝑥 = 𝑐 if

  • A 𝑐 = 2 , 𝑐 = 1
  • B 𝑐 = 1 , 𝑐 = 2
  • C 𝑐 = 1 , 𝑐 = 2
  • D 𝑐 = 1 , 𝑐 = 2
  • E 𝑐 = 2 , 𝑐 = 2

Q9:

Find the set of values of 𝑥 for which the function given by 𝑓 ( 𝑥 ) = 5 𝑥 1 3 is continuous.

  • A 𝑥
  • B 𝑥 1 5
  • C 𝑥
  • D 𝑥

Q10:

Find the set on which 𝑓 ( 𝑥 ) = 𝑥 2 2 𝑥 2 𝑥 6 3 2 is continuous.

  • A 𝑓 ( 𝑥 ) is continuous on .
  • B 𝑓 ( 𝑥 ) is continuous on { 2 2 } .
  • C 𝑓 ( 𝑥 ) is continuous on { 9 , 7 } .
  • D 𝑓 ( 𝑥 ) is continuous on { 9 , 7 } .

Q11:

Use continuity to evaluate l i m s i n s i n 𝑥 𝜋 6 ( 2 𝑥 7 6 𝑥 ) .

  • A2
  • B0
  • C 7
  • D 3 2
  • E6

Q12:

Find the set on which 𝑓 ( 𝑥 ) = 𝑥 + 9 𝑥 + 1 c o s is continuous.

  • AThe function 𝑓 is continuous on 𝑥 𝑥 = 3 𝜋 2 + 2 𝑛 𝜋 , 𝑛 .
  • BThe function 𝑓 is continuous on { 𝑥 𝑥 = 2 𝜋 𝑛 , 𝑛 } .
  • CThe function 𝑓 is continuous on 𝑥 𝑥 = 𝜋 2 + 𝑛 𝜋 , 𝑛 .
  • DThe function 𝑓 is continuous on { 𝑥 𝑥 = 𝜋 + 2 𝜋 𝑛 , 𝑛 } .
  • EThe function 𝑓 is continuous on because 𝑥 + 9 and c o s 𝑥 + 1 are continuous on .

Q13:

Given that 𝑓 and 𝑔 are continuous functions such that 𝑔 ( 6 ) = 6 and l i m 𝑥 6 [ 𝑓 ( 𝑥 ) 9 𝑓 ( 𝑥 ) 𝑔 ( 𝑥 ) ] = 5 , determine 𝑓 ( 6 ) .

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

Q14:

Which of the following holds for 𝑓 ( 𝑥 ) = ( 7 𝑥 + 3 ) + 3 5 𝑥 7 s i n ?

  • A The function 𝑓 is continuous on because 3 5 𝑥 s i n is continuous on .
  • B The function 𝑓 is continuous on because ( 7 𝑥 + 3 ) 7 is continuous on .
  • C The function 𝑓 is continuous on 𝑥 𝑥 = 𝜋 2 + 𝑛 𝜋 , 𝑛 .
  • D The function 𝑓 is continuous on because both of ( 7 𝑥 + 3 ) 7 and 3 5 𝑥 s i n are continuous on .
  • E The function 𝑓 is continuous on { 𝑥 𝑥 = 𝜋 𝑛 , 𝑛 } .

Q15:

Use continuity to evaluate l i m 2 𝑥 2 2 + 3 𝑥 .

  • A 2 7 0
  • B 4 7 0
  • C 7 0
  • D 8 7 0
  • E8

Q16:

Use continuity to evaluate l i m 𝑥 1 3 𝑥 + 2 𝑥 1 5 2 .

Q17:

Use continuity to evaluate l i m l n 𝑥 5 2 5 + 2 𝑥 6 𝑥 .

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

Q18:

Determine the value of 𝑎 that makes the function 𝑓 continuous at 𝑥 = 9 , given

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

Q19:

Suppose What can be said of the continuity of 𝑓 at 𝑥 = 0 ?

  • A The function is discontinuous at 𝑥 = 0 because l i m 𝑥 0 𝑓 ( 𝑥 ) does not exist.
  • B The function is continuous on .
  • C The function is discontinuous at 𝑥 = 0 because 𝑓 ( 0 ) is undefined.
  • D The function 𝑓 is continuous at 𝑥 = 0 .
  • E The function is discontinuous at 𝑥 = 0 because 𝑓 ( 0 ) 𝑓 ( 𝑥 ) l i m 𝑥 0 .

Q20:

Given 𝑓 ( 𝑥 ) = 𝑥 1 0 0 𝑥 1 0 , if possible or necessary, define 𝑓 ( 1 0 ) so that 𝑓 is continuous at 𝑥 = 1 0 .

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

Q21:

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 𝑥 = 6 .
  • BThe function 𝑓 can be defined to be continuous at 𝑥 = 6 as
  • CThe function 𝑓 cannot be defined to be continuous at 𝑥 = 6 because l i m 𝑥 6 𝑓 ( 𝑥 ) does not exist.
  • DThe function 𝑓 can be defined to be continuous at 𝑥 = 6 as
  • EThe function 𝑓 cannot be defined to be continuous at 𝑥 = 6 because 𝑓 ( 6 ) is undefined.

Q22:

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

  • Acontinuous
  • Bdiscontinuous

Q23:

Given 𝑓 ( 𝑥 ) = 9 𝑥 + 𝑥 2 2 , what can be said of the continuity of 𝑓 at 𝑥 = 7 ?

  • AThe function is discontinuous at 𝑥 = 7 because l i m 𝑥 7 𝑓 ( 𝑥 ) does not exist.
  • BThe function is continuous on .
  • CThe function is discontinuous at 𝑥 = 7 because 𝑓 ( 7 ) is undefined.
  • DThe function is continuous at 𝑥 = 7 .
  • EThe function is discontinuous at 𝑥 = 7 because 𝑓 ( 7 ) 𝑓 ( 𝑥 ) . l i m 𝑥 7

Q24:

Find the set on which 𝑓 ( 𝑥 ) = 𝑥 + 5 ( 𝑥 + 8 ) s i n c o s is continuous.

  • A 𝑓 ( 𝑥 ) is continuous on because 𝑥 𝑥 s i n is continuous on .
  • B 𝑓 ( 𝑥 ) is continuous on { 8 } .
  • C 𝑓 ( 𝑥 ) is continuous on because 𝑥 ( 𝑥 + 8 ) c o s is continuous on .
  • D 𝑓 ( 𝑥 ) is continuous on because 𝑥 𝑥 s i n and 𝑥 ( 𝑥 + 8 ) c o s are continuous on .

Q25:

Discuss the continuity of the function 𝑓 ( 𝑥 ) = 3 𝑥 2 3 at 𝑥 = 6 .

  • A The function is discontinuous at 𝑥 = 6 because l i m 𝑥 6 𝑓 ( 𝑥 ) does not exist.
  • B The function is continuous on .
  • C The function is discontinuous at 𝑥 = 6 because 𝑓 ( 6 ) is undefined.
  • D The function is continuous at 𝑥 = 6 .
  • E The function is discontinuous at 𝑥 = 6 because 𝑓 ( 6 ) 𝑓 ( 𝑥 ) l i m 𝑥 6 .

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