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)=1𝑥𝑥<0,0𝑥=0,1+2𝑥𝑥>0.whenwhenwhen

What is 𝑓(0)?

What is lim𝑓(𝑥)?

What is lim𝑓(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 removable discontinuity at 𝑥=0.
  • CThe function 𝑓 has a jump discontinuity at 𝑥=0.
  • DThe function 𝑓 does not have a discontinuity at 𝑥=0.

Q2:

Consider the function 𝑓(𝑥)=1𝑥,𝑥<0,0,𝑥=0,1𝑥,𝑥>0.sinwhenwhenwhen

What is 𝑓(0)?

What is lim𝑓(𝑥)?

  • A1
  • BThe limit does not exist.
  • C0
  • D
  • E+

What is lim𝑓(𝑥)?

  • A
  • B1
  • CThe limit does not exist.
  • D+
  • E0

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

Q3:

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

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

Q4:

Consider the function 𝑓(𝑥)=𝑥1𝑥2,3𝑥1𝑥>2. Find the type of discontinuity that the function 𝑓 has at 𝑥=2, if it has any.

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

Q5:

Consider the function 𝑓(𝑥)=𝑥9𝑥3.

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.

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 does not have a discontinuity at 𝑥=0.
  • BThe function 𝑓 has a removable discontinuity at 𝑥=0.
  • CThe function 𝑓 has an infinite discontinuity at 𝑥=0.
  • DThe function 𝑓 has a jump 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 an infinite 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 a jump 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 an infinite discontinuity at 𝑥=6.
  • BThe function 𝑓 has a jump discontinuity at 𝑥=6.
  • CThe function 𝑓 has a removable discontinuity at 𝑥=6.
  • DThe function 𝑓 does not have a discontinuity at 𝑥=6.

Q7:

Consider the function 𝑓(𝑥)=6𝑥8𝑥+23𝑥1𝑥13,43𝑥=13. Find the type of discontinuity that the function 𝑓 has at 𝑥=13, if it has any.

  • AThe function 𝑓 does not have a discontinuity at 𝑥=13.
  • BThe function 𝑓 has a removable discontinuity at 𝑥=13.
  • CThe function 𝑓 has a jump discontinuity at 𝑥=13.
  • DThe function 𝑓 has an infinite discontinuity at 𝑥=13.

Q8:

Consider the function 𝑓(𝑥)=𝑥+3𝑥1. Find the type of discontinuity that the function 𝑓 has at 𝑥=1, if it has any.

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

Q10:

Consider the function 𝑓(𝑥)=𝑥+1𝑥+5𝑥+4.

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

  • AThe function 𝑓 has a jump discontinuity at 𝑥=4.
  • BThe function 𝑓 has an infinite discontinuity at 𝑥=4.
  • CThe function 𝑓 does not have a discontinuity at 𝑥=4.
  • DThe function 𝑓 has a removable 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 a jump discontinuity at 𝑥=1.
  • BThe function 𝑓 has a removable discontinuity at 𝑥=1.
  • CThe function 𝑓 does not have a discontinuity at 𝑥=1.
  • DThe function 𝑓 has an infinite discontinuity at 𝑥=1.

Q11:

Consider the function 𝑓(𝑥)=2𝑥cot. Find the type of discontinuity that the function 𝑓 has at 𝑥=𝜋2, if it has any.

  • AThe function 𝑓 does not have a discontinuity at 𝑥=𝜋2.
  • BThe function 𝑓 has a jump discontinuity at 𝑥=𝜋2.
  • CThe function 𝑓 has an infinite discontinuity at 𝑥=𝜋2.
  • DThe function 𝑓 has a removable discontinuity at 𝑥=𝜋2.

Q12:

Consider the function 𝑓(𝑥)=2𝑥3𝑥2𝑥2. 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 an infinite discontinuity at 𝑥=2.
  • CThe function 𝑓 does not have a discontinuity at 𝑥=2.
  • DThe function 𝑓 has a removable discontinuity at 𝑥=2.

Q13:

Consider the function 𝑓(𝑥)=𝑥𝑥𝑥𝜋2,𝑥𝑥𝑥>𝜋2.coscot What type of discontinuity does the function 𝑓 have at 𝑥=𝜋2, if it has any?

  • AThe function 𝑓 does not have a discontinuity at 𝑥=𝜋2.
  • BThe function 𝑓 has an infinite discontinuity at 𝑥=𝜋2.
  • CThe function 𝑓 has a removable discontinuity at 𝑥=𝜋2.
  • DThe function 𝑓 has a jump discontinuity at 𝑥=𝜋2.

Q14:

Given 𝑓(𝑥)=𝑥16𝑥4, if possible, define 𝑓(𝑥) so that 𝑓 is continuous at 𝑥=4.

  • AThe function is already continuous at 𝑥=4.
  • BNo value of 𝑓(𝑥) will make 𝑓 continuous because lim𝑓(𝑥) does not exist.
  • C𝑓(𝑥)=4 makes 𝑓 continuous at 𝑥=4.
  • DThe function cannot be made continuous at 𝑥=4 because 𝑓(𝑥) is undefined.

Q15:

Given 𝑓(𝑥)=1𝑥79𝑥5𝑥14, if possible, define 𝑓(𝑥) so that 𝑓 is continuous at 𝑥=7.

  • AThe function is already continuous at 𝑥=7.
  • BNo value of 𝑓(𝑥) will make 𝑓 continuous because lim𝑓(𝑥) does not exist.
  • C𝑓(𝑥)=19 makes 𝑓 continuous at 𝑥=7.
  • DThe function cannot be made continuous at 𝑥=7 because 𝑓(𝑥) is undefined.

Q16:

Suppose 𝑓(𝑥)=𝑥3𝑥<0,𝑥𝑥6𝑥𝑥>0.cosiftansinif If possible, define 𝑓(𝑥) so that 𝑓 is continuous at 𝑥=3.

  • AThe function can not be made continuous at 𝑥=3 because lim𝑓(𝑥) does not exist.
  • B𝑓(𝑥)=0 would make 𝑓 continuous at 𝑥=3.
  • C𝑓(𝑥)=13 would make 𝑓 continuous at 𝑥=3.
  • DThe function is already continuous at 𝑥=3.

Q17:

Given 𝑓(𝑥)=|𝑥5|𝑥125, if possible, define 𝑓(𝑥) so that 𝑓 is continuous at 𝑥=3.

  • AThe function is already continuous at 𝑥=3.
  • B𝑓(3)=125 would make 𝑓 continuous at 𝑥=3
  • CThe function can not be made continuous at 𝑥=3 because lim𝑓(𝑥) does not exist.
  • D𝑓(3)=0 would make 𝑓 continuous at 𝑥=3.

Q18:

Given 𝑓(𝑥)=6𝑥27𝑥+276𝑥9, if possible or necessary, define 𝑓(32) so that 𝑓 is continuous at 𝑥=32.

  • ANo value of 𝑓(32) will make 𝑓 continuous because lim𝑓(𝑥) does not exist.
  • B𝑓(32)=32 makes 𝑓 continuous at 𝑥=32.
  • CThe function cannot be made continuous at 𝑥=32 because 𝑓(32) is undefined.
  • DThe function is already continuous at 𝑥=32.

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