Lesson Worksheet: Continuity at a Point Mathematics • 12th Grade

In this worksheet, we will practice checking the continuity of a function at a given point.

Q1:

Discuss the continuity of the function 𝑓 at 𝑥=𝜋2, given 𝑓(𝑥)=7𝑥+7𝑥,𝑥𝜋2,62𝑥1,𝑥>𝜋2.sincoscos

  • AThe function is discontinuous at 𝑥=𝜋2 because 𝑓𝜋2 is undefined.
  • BThe function is continuous at 𝑥=𝜋2.
  • CThe function is discontinuous on .
  • DThe function is discontinuous at 𝑥=𝜋2 because lim𝑓(𝑥) does not exist.
  • EThe function is discontinuous at 𝑥=𝜋2 because lim𝑓(𝑥)𝑓𝜋2.

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 .
  • 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 lim𝑓(𝑥) does not exist.

Q3:

Given 𝑓(𝑥)=𝑥+𝑥2𝑥1, 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 lim𝑓(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.

Q4:

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

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

Q5:

Is 𝑓(𝑥)=2𝑥+4𝑥+2𝑥<2,0𝑥=2,𝑥+6𝑥+8𝑥+2𝑥>2ififif continuous at 𝑥=2?

  • AYes
  • BNo

Q6:

Given 𝑓(𝑥)=4+𝑥28𝑥, define, if possible, 𝑓(0) so that 𝑓 is continuous at 𝑥=0.

  • AThe function 𝑓 cannot be made continuous at 𝑥=0 because lim𝑓(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.

Q7:

Discuss the continuity of the function 𝑓 at 𝑥=0, given that 𝑓(𝑥)=6𝑥𝑥𝑥4𝑥,𝑥<0,𝑥+5𝑥+4,𝑥0.sintan

  • AThe function is discontinuous on .
  • BThe function is discontinuous at 𝑥=0 because lim𝑓(𝑥) does not exist.
  • CThe function is continuous at 𝑥=0.
  • DThe function is discontinuous at 𝑥=0 because 𝑓(0) is undefined.
  • EThe function is discontinuous at 𝑥=0 because lim𝑓(𝑥)𝑓(0).

Q8:

Find the value of 𝑘 which makes the function 𝑓 continuous at 𝑥=3, given 𝑓(𝑥)=𝑥3𝑥3𝑥3,𝑘𝑥=3.ifif

  • A127
  • B54
  • C154
  • D127
  • E227

Q9:

Find the values of 𝑎 and 𝑏 that make the function 𝑓 continuous at 𝑥=2 and at 𝑥=2, given 𝑓(𝑥)=3𝑥5,𝑥2,𝑎𝑥+𝑏,2<𝑥<2,2𝑥3,𝑥2.

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

Q10:

Find the values of 𝑎 and 𝑏 that make the function 𝑓 continuous at 𝑥=1 and 𝑥=6, given that 𝑓(𝑥)=3𝑥+11,𝑥6,𝑎𝑥+𝑏,6<𝑥<1,5𝑥+10,𝑥1.

  • A𝑎=125, 𝑏=375
  • B𝑎=375, 𝑏=125
  • C𝑎=617, 𝑏=127
  • D𝑎=127, 𝑏=617
  • E𝑎=125, 𝑏=2575

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