Worksheet: Classifying Discontinuities

In this worksheet, we will practice differentiating between the three types of function discontinuity at a given point.

Q1:

Consider the function 𝑓 ( 𝑥 ) = 1 𝑥 𝑥 < 0 , 0 𝑥 = 0 , 1 + 2 𝑥 𝑥 > 0 . w h e n w h e n w h e n

What is 𝑓 ( 0 ) ?

What is l i m 𝑓 ( 𝑥 ) ?

What is l i m 𝑓 ( 𝑥 ) ?

What type of discontinuity does the function 𝑓 have at 𝑥 = 0 ?

  • AThe function 𝑓 has an essential discontinuity at 𝑥 = 0 .
  • BThe function 𝑓 has a jump discontinuity at 𝑥 = 0 .
  • CThe function 𝑓 does not have a discontinuity at 𝑥 = 0 .
  • DThe function 𝑓 has a removable discontinuity at 𝑥 = 0 .

Q2:

Consider the function

What is 𝑓 ( 0 ) ?

What is l i m 𝑥 0 𝑓 ( 𝑥 ) ?

  • A1
  • B0
  • CThe limit does not exist.
  • D
  • E +

What is l i m 𝑥 0 + 𝑓 ( 𝑥 ) ?

  • A
  • BThe limit does not exist.
  • C0
  • D1
  • E +

What type of discontinuity does the function 𝑓 have at 𝑥 = 0 ?

  • AThe function 𝑓 does not have a discontinuity at 𝑥 = 0 .
  • BThe function 𝑓 has an essential discontinuity at 𝑥 = 0 .
  • CThe function 𝑓 has a removable discontinuity at 𝑥 = 0 .
  • DThe function 𝑓 has a jump discontinuity at 𝑥 = 0 .

Q3:

Consider the function 𝑓 ( 𝑥 ) = 𝑥 + 𝑥 𝑥 s i n c o s c o t . Find the type of discontinuity that the function 𝑓 has at 𝑥 = 𝜋 , if it has any.

  • AThe function 𝑓 has a removable discontinuity at 𝑥 = 𝜋 .
  • BThe function 𝑓 does not have a discontinuity at 𝑥 = 𝜋 .
  • CThe function 𝑓 has a jump discontinuity at 𝑥 = 𝜋 .
  • DThe function 𝑓 has an infinite discontinuity at 𝑥 = 𝜋 .

Q4:

Consider the function 𝑓 ( 𝑥 ) = 𝑥 1 𝑥 2 , 3 𝑥 1 𝑥 > 2 . Find the type of discontinuity that the function 𝑓 has at 𝑥 = 2 , if it has any.

  • AThe function 𝑓 has a jump discontinuity at 𝑥 = 2 .
  • BThe function 𝑓 has a removable discontinuity at 𝑥 = 2 .
  • CThe function 𝑓 does not have a discontinuity at 𝑥 = 2 .
  • DThe function 𝑓 has an infinite discontinuity at 𝑥 = 2 .

Q5:

Consider the function 𝑓 ( 𝑥 ) = 𝑥 9 𝑥 3 .

Find the type of discontinuity that the function 𝑓 has at 𝑥 = 3 , if it has any.

  • AThe function 𝑓 has a removable discontinuity at 𝑥 = 3 .
  • BThe function 𝑓 has a jump discontinuity at 𝑥 = 3 .
  • CThe function 𝑓 has an infinite discontinuity at 𝑥 = 3 .
  • DThe function 𝑓 does not have a discontinuity at 𝑥 = 3 .

Q6:

Consider the graph of the function 𝑦 = 𝑓 ( 𝑥 ) shown.

Find the type of discontinuity that the function 𝑓 has at 𝑥 = 0 , if it has any.

  • AThe function 𝑓 has a removable discontinuity at 𝑥 = 0 .
  • BThe function 𝑓 has a jump discontinuity at 𝑥 = 0 .
  • CThe function 𝑓 does not have a discontinuity at 𝑥 = 0 .
  • DThe function 𝑓 has an infinite discontinuity at 𝑥 = 0 .

Find the type of discontinuity that the function 𝑓 has at 𝑥 = 2 , if it has any.

  • AThe function 𝑓 has a jump discontinuity at 𝑥 = 2 .
  • BThe function 𝑓 has a removable discontinuity at 𝑥 = 2 .
  • CThe function 𝑓 does not have a discontinuity at 𝑥 = 2 .
  • DThe function 𝑓 has an infinite discontinuity at 𝑥 = 2 .

Find the type of discontinuity that the function 𝑓 has at 𝑥 = 5 , if it has any.

  • AThe function 𝑓 has a removable discontinuity at 𝑥 = 5 .
  • BThe function 𝑓 has a jump discontinuity at 𝑥 = 5 .
  • CThe function 𝑓 has an infinite discontinuity at 𝑥 = 5 .
  • DThe function 𝑓 does not have a discontinuity at 𝑥 = 5 .

Find the type of discontinuity that the function 𝑓 has at 𝑥 = 6 , if it has any.

  • AThe function 𝑓 has a jump discontinuity at 𝑥 = 6 .
  • BThe function 𝑓 has a removable discontinuity at 𝑥 = 6 .
  • CThe function 𝑓 has an infinite discontinuity at 𝑥 = 6 .
  • DThe function 𝑓 does not have a discontinuity at 𝑥 = 6 .

Q7:

Consider the function 𝑓 ( 𝑥 ) = 6 𝑥 8 𝑥 + 2 3 𝑥 1 𝑥 1 3 , 4 3 𝑥 = 1 3 . Find the type of discontinuity that the function 𝑓 has at 𝑥 = 1 3 , if it has any.

  • AThe function 𝑓 has a jump discontinuity at 𝑥 = 1 3 .
  • BThe function 𝑓 has an infinite discontinuity at 𝑥 = 1 3 .
  • CThe function 𝑓 does not have a discontinuity at 𝑥 = 1 3 .
  • DThe function 𝑓 has a removable discontinuity at 𝑥 = 1 3 .

Q8:

Consider the function 𝑓 ( 𝑥 ) = 𝑥 + 3 𝑥 1 . Find the type of discontinuity that the function 𝑓 has at 𝑥 = 1 , if it has any.

  • AThe function 𝑓 has an infinite discontinuity at 𝑥 = 1 .
  • BThe function 𝑓 does not have a discontinuity at 𝑥 = 1 .
  • CThe function 𝑓 has a jump discontinuity at 𝑥 = 1 .
  • DThe function 𝑓 has a removable discontinuity at 𝑥 = 1 .

Q9:

Consider the function 𝑓 ( 𝑥 ) = | 𝑥 + 3 | 𝑥 + 3 . Find the type of discontinuity that the function 𝑓 has at 𝑥 = 3 , if it has any.

  • AThe function 𝑓 does not have a discontinuity at 𝑥 = 3 .
  • BThe function 𝑓 has a jump discontinuity at 𝑥 = 3 .
  • CThe function 𝑓 has an infinite discontinuity at 𝑥 = 3 .
  • DThe function 𝑓 has a removable discontinuity at 𝑥 = 3 .

Q10:

Consider the function 𝑓 ( 𝑥 ) = 𝑥 + 1 𝑥 + 5 𝑥 + 4 .

Find the type of discontinuity that the function 𝑓 has at 𝑥 = 4 , if it has any.

  • AThe function 𝑓 has a removable discontinuity at 𝑥 = 4 .
  • BThe function 𝑓 has a jump discontinuity at 𝑥 = 4 .
  • CThe function 𝑓 has an infinite discontinuity at 𝑥 = 4 .
  • DThe function 𝑓 does not have a discontinuity at 𝑥 = 4 .

Find the type of discontinuity that the function 𝑓 has at 𝑥 = 1 , if it has any.

  • AThe function 𝑓 has an infinite discontinuity at 𝑥 = 1 .
  • BThe function 𝑓 does not have a discontinuity at 𝑥 = 1 .
  • CThe function 𝑓 has a removable discontinuity at 𝑥 = 1 .
  • DThe function 𝑓 has a jump discontinuity at 𝑥 = 1 .

Q11:

Consider the function 𝑓 ( 𝑥 ) = 2 𝑥 c o t . Find the type of discontinuity that the function 𝑓 has at 𝑥 = 𝜋 2 , if it has any.

  • AThe function 𝑓 does not have a discontinuity at 𝑥 = 𝜋 2 .
  • BThe function 𝑓 has a removable discontinuity at 𝑥 = 𝜋 2 .
  • CThe function 𝑓 has an infinite discontinuity at 𝑥 = 𝜋 2 .
  • DThe function 𝑓 has a jump discontinuity at 𝑥 = 𝜋 2 .

Q12:

Consider the function 𝑓 ( 𝑥 ) = 2 𝑥 3 𝑥 2 𝑥 2 . Find the type of discontinuity that the function 𝑓 has at 𝑥 = 2 , if it has any.

  • AThe function 𝑓 has a jump discontinuity at 𝑥 = 2 .
  • BThe function 𝑓 does not have a discontinuity at 𝑥 = 2 .
  • CThe function 𝑓 has an infinite discontinuity at 𝑥 = 2 .
  • DThe function 𝑓 has a removable discontinuity at 𝑥 = 2 .

Q13:

Consider the function 𝑓 ( 𝑥 ) = 𝑥 𝑥 𝑥 𝜋 2 , 𝑥 𝑥 𝑥 > 𝜋 2 . c o s c o t What type of discontinuity does the function 𝑓 have at 𝑥 = 𝜋 2 , if it has any?

  • AThe function 𝑓 does not have a discontinuity at 𝑥 = 𝜋 2 .
  • BThe function 𝑓 has a removable discontinuity at 𝑥 = 𝜋 2 .
  • CThe function 𝑓 has a jump discontinuity at 𝑥 = 𝜋 2 .
  • DThe function 𝑓 has an infinite discontinuity at 𝑥 = 𝜋 2 .

Q14:

Given 𝑓 ( 𝑥 ) = 𝑥 1 6 𝑥 4 , if possible, define 𝑓 ( 𝑥 ) so that 𝑓 is continuous at 𝑥 = 4 .

  • AThe function is already continuous at 𝑥 = 4 .
  • B 𝑓 ( 𝑥 ) = 4 makes 𝑓 continuous at 𝑥 = 4 .
  • CThe function cannot be made continuous at 𝑥 = 4 because 𝑓 ( 𝑥 ) is undefined.
  • DNo value of 𝑓 ( 𝑥 ) will make 𝑓 continuous because l i m 𝑓 ( 𝑥 ) does not exist.

Q15:

Given 𝑓 ( 𝑥 ) = 1 𝑥 7 9 𝑥 5 𝑥 1 4 , if possible, define 𝑓 ( 𝑥 ) so that 𝑓 is continuous at 𝑥 = 7 .

  • AThe function is already continuous at 𝑥 = 7 .
  • B 𝑓 ( 𝑥 ) = 1 9 makes 𝑓 continuous at 𝑥 = 7 .
  • CNo value of 𝑓 ( 𝑥 ) will make 𝑓 continuous because l i m 𝑓 ( 𝑥 ) does not exist.
  • DThe function cannot be made continuous at 𝑥 = 7 because 𝑓 ( 𝑥 ) is undefined.

Q16:

Suppose 𝑓 ( 𝑥 ) = 𝑥 3 𝑥 < 0 , 𝑥 𝑥 6 𝑥 𝑥 > 0 . c o s i f t a n s i n i f If possible, define 𝑓 ( 𝑥 ) so that 𝑓 is continuous at 𝑥 = 3 .

  • AThe function is already continuous at 𝑥 = 3 .
  • B 𝑓 ( 𝑥 ) = 0 would make 𝑓 continuous at 𝑥 = 3 .
  • C 𝑓 ( 𝑥 ) = 1 3 would make 𝑓 continuous at 𝑥 = 3 .
  • DThe function can not be made continuous at 𝑥 = 3 because l i m 𝑓 ( 𝑥 ) does not exist.

Q17:

Given 𝑓 ( 𝑥 ) = | 𝑥 5 | 𝑥 1 2 5 , if possible, define 𝑓 ( 𝑥 ) so that 𝑓 is continuous at 𝑥 = 3 .

  • AThe function is already continuous at 𝑥 = 3 .
  • B 𝑓 ( 3 ) = 0 would make 𝑓 continuous at 𝑥 = 3 .
  • C 𝑓 ( 3 ) = 1 2 5 would make 𝑓 continuous at 𝑥 = 3
  • DThe function can not be made continuous at 𝑥 = 3 because l i m 𝑓 ( 𝑥 ) does not exist.

Q18:

Given 𝑓 ( 𝑥 ) = 6 𝑥 2 7 𝑥 + 2 7 6 𝑥 9 , if possible or necessary, define 𝑓 ( 3 2 ) so that 𝑓 is continuous at 𝑥 = 3 2 .

  • ANo value of 𝑓 ( 3 2 ) will make 𝑓 continuous because l i m 𝑓 ( 𝑥 ) does not exist.
  • B 𝑓 ( 3 2 ) = 3 2 makes 𝑓 continuous at 𝑥 = 3 2 .
  • CThe function cannot be made continuous at 𝑥 = 3 2 because 𝑓 ( 3 2 ) is undefined.
  • DThe function is already continuous at 𝑥 = 3 2 .

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