Lesson Worksheet: Continuity and Limits Mathematics

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In this worksheet, we will practice checking the continuity of a function at a given point by using limits and using the mean value theorem on continuous functions.

Q1:

Given 𝑓(𝑥)=𝑥+𝑥−2𝑥−1, if possible or necessary, define 𝑓(1) so that 𝑓 is continuous at 𝑥=1.

  • AThe function is already continuous at 𝑥=1.
  • BNo value of 𝑓(1) will make 𝑓 continuous because lim→𝑓(𝑥) does not exist.
  • C𝑓(1)=3 makes 𝑓 continuous at 𝑥=1.
  • DThe function cannot be made continuous at 𝑥=1 because 𝑓(1) is undefined.

Q2:

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

  • A𝑓(0)=1 will make 𝑓 continuous at 𝑥=0.
  • B𝑓 is already continuous on ℝ.
  • CThe function is already continuous at 𝑥=0.
  • D𝑓(0)=0 will make 𝑓 continuous at 𝑥=0.
  • EThe function cannot be made continuous at 𝑥=0 by defining 𝑓(0) as lim→𝑓(𝑥) does not exist.

Q3:

Find the values of 𝑎 and 𝑏 that make the function 𝑓 continuous at 𝑥=−2 and at 𝑥=2, given 𝑓(𝑥)=3𝑥−5,𝑥≤−2,𝑎𝑥+𝑏,−2<𝑥<2,2𝑥−3,𝑥≥2.

  • A𝑎=2, 𝑏=−5
  • B𝑎=−11, 𝑏=−5
  • C𝑎=6, 𝑏=−1
  • D𝑎=4, 𝑏=−3

Q4:

Find the value of 𝑘 which makes the function 𝑓 continuous at 𝑥=3, given 𝑓(𝑥)=𝑥−3𝑥−3𝑥≠3,𝑘𝑥=3.ifif

  • A−127
  • B−54
  • C−154
  • D127
  • E−227

Q5:

Given 𝑓(𝑥)=√4+𝑥−28𝑥, define, if possible, 𝑓(0) so that 𝑓 is continuous at 𝑥=0.

  • AThe function 𝑓 cannot be made continuous at 𝑥=0 because lim→𝑓(𝑥) does not exist.
  • BSetting 𝑓(0)=132 makes 𝑓 continuous at 𝑥=0.
  • CThe function 𝑓 cannot be made continuous at 𝑥=0 because 𝑓(0) is undefined.
  • DThe function is already continuous at 𝑥=0.

Q6:

Find the values of 𝑐 which make the function 𝑓 continuous at 𝑥=𝑐 if 𝑓(𝑥)=2+𝑥𝑥≤𝑐,−3𝑥𝑥>𝑐.ifif

  • A𝑐=−2, 𝑐=1
  • B𝑐=−1, 𝑐=2
  • C𝑐=−1, 𝑐=−2
  • D𝑐=2, 𝑐=2
  • E𝑐=1, 𝑐=2

Q7:

Determine the value of 𝑎 that makes the function 𝑓 continuous at 𝑥=5, given 𝑓(𝑥)=2𝑥+𝑥−5,𝑥≠5,−2𝑎,𝑥=5.

Q8:

Setting 𝑓(𝑎)=54 and 𝑓(𝑥)=𝑥−𝑎𝑥−𝑎 when 𝑥≠𝑎 makes 𝑓 continuous at 𝑥=𝑎. Determine 𝑎.

  • A12
  • B3
  • C13
  • D2

Q9:

Discuss the continuity of the function 𝑓 at 𝑥=5 given 𝑓(𝑥)=⎧⎪⎨⎪⎩8𝑥+1𝑥≤5,𝑥−25𝑥−125𝑥>5.ifif

  • AThe function is discontinuous at 𝑥=5 because lim→𝑓(𝑥) does not exist.
  • BThe function is discontinuous at 𝑥=5 because 𝑓(5) is undefined.
  • CThe function is discontinuous at 𝑥=5 because lim→𝑓(𝑥)≠𝑓(5).
  • DThe function is continuous at 𝑥=5.

Q10:

Given 𝑓(𝑥)=−7𝑥+8𝑥<−8,𝑥+2𝑥+4𝑥>−8.ifif If possible or necessary, define 𝑓(−8) so that 𝑓 is continuous at 𝑥=−8.

  • AThe function is already continuous at 𝑥=−8.
  • B𝑓(−8)=0 would make 𝑓 continuous at 𝑥=−8.
  • C𝑓(−8)=6 would make 𝑓 continuous at 𝑥=−8.
  • DThe function cannot be made continuous at 𝑥=−8 because lim→𝑓(𝑥)≠lim→𝑓(𝑥).

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