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 π‘₯=πœ‹2, given 𝑓(π‘₯)=ο΄βˆ’7π‘₯+7π‘₯,π‘₯β‰€πœ‹2,62π‘₯βˆ’1,π‘₯>πœ‹2.sincoscos

  • AThe function is discontinuous at π‘₯=πœ‹2 because π‘“ο€»πœ‹2 is undefined.
  • BThe function is continuous at π‘₯=πœ‹2.
  • CThe function is discontinuous on ℝ.
  • DThe function is discontinuous at π‘₯=πœ‹2 because limο—β†’ο‘½οŽ‘π‘“(π‘₯) does not exist.
  • EThe function is discontinuous at π‘₯=πœ‹2 because limο—β†’ο‘½οŽ‘π‘“(π‘₯)β‰ π‘“ο€»πœ‹2.

Q2:

Given 𝑓(π‘₯)=1π‘₯, if possible or necessary, define 𝑓(0) so that 𝑓 is continuous at π‘₯=0.

  • A𝑓(0)=1 will make 𝑓 continuous at π‘₯=0.
  • B𝑓 is already continuous on ℝ.
  • CThe function is already continuous at π‘₯=0.
  • D𝑓(0)=0 will make 𝑓 continuous at π‘₯=0.
  • EThe function cannot be made continuous at π‘₯=0 by defining 𝑓(0) as limο—β†’οŠ¦π‘“(π‘₯) does not exist.

Q3:

Given 𝑓(π‘₯)=π‘₯+π‘₯βˆ’2π‘₯βˆ’1, if possible or necessary, define 𝑓(1) so that 𝑓 is continuous at π‘₯=1.

  • AThe function is already continuous at π‘₯=1.
  • BNo value of 𝑓(1) will make 𝑓 continuous because limο—β†’οŠ§π‘“(π‘₯) does not exist.
  • C𝑓(1)=3 makes 𝑓 continuous at π‘₯=1.
  • DThe function cannot be made continuous at π‘₯=1 because 𝑓(1) is undefined.

Q4:

Find the values of 𝑐 which make the function 𝑓 continuous at π‘₯=𝑐 if 𝑓(π‘₯)=2+π‘₯π‘₯≀𝑐,βˆ’3π‘₯π‘₯>𝑐.ifif

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

Q5:

Is 𝑓(π‘₯)=⎧βŽͺ⎨βŽͺ⎩2π‘₯+4π‘₯+2π‘₯<βˆ’2,0π‘₯=βˆ’2,π‘₯+6π‘₯+8π‘₯+2π‘₯>βˆ’2ififif continuous at π‘₯=βˆ’2?

  • AYes
  • BNo

Q6:

Given 𝑓(π‘₯)=√4+π‘₯βˆ’28π‘₯, define, if possible, 𝑓(0) so that 𝑓 is continuous at π‘₯=0.

  • AThe function 𝑓 cannot be made continuous at π‘₯=0 because limο—β†’οŠ¦π‘“(π‘₯) does not exist.
  • BSetting 𝑓(0)=132 makes 𝑓 continuous at π‘₯=0.
  • CThe function 𝑓 cannot be made continuous at π‘₯=0 because 𝑓(0) is undefined.
  • DThe function is already continuous at π‘₯=0.

Q7:

Discuss the continuity of the function 𝑓 at π‘₯=0, given that 𝑓(π‘₯)=⎧βŽͺ⎨βŽͺ⎩6π‘₯βˆ’π‘₯π‘₯4π‘₯,π‘₯<0,π‘₯+5π‘₯+4,π‘₯β‰₯0.sintan

  • AThe function is discontinuous on ℝ.
  • BThe function is discontinuous at π‘₯=0 because limο—β†’οŠ¦π‘“(π‘₯) does not exist.
  • CThe function is continuous at π‘₯=0.
  • DThe function is discontinuous at π‘₯=0 because 𝑓(0) is undefined.
  • EThe function is discontinuous at π‘₯=0 because limο—β†’οŠ¦π‘“(π‘₯)≠𝑓(0).

Q8:

Find the value of π‘˜ which makes the function 𝑓 continuous at π‘₯=3, given 𝑓(π‘₯)=π‘₯βˆ’3π‘₯βˆ’3π‘₯β‰ 3,π‘˜π‘₯=3.ifif

  • Aβˆ’127
  • Bβˆ’54
  • Cβˆ’154
  • D127
  • Eβˆ’227

Q9:

Find the values of π‘Ž and 𝑏 that make the function 𝑓 continuous at π‘₯=βˆ’2 and at π‘₯=2, given 𝑓(π‘₯)=3π‘₯βˆ’5,π‘₯β‰€βˆ’2,π‘Žπ‘₯+𝑏,βˆ’2<π‘₯<2,2π‘₯βˆ’3,π‘₯β‰₯2.

  • Aπ‘Ž=2, 𝑏=βˆ’5
  • Bπ‘Ž=βˆ’11, 𝑏=βˆ’5
  • Cπ‘Ž=6, 𝑏=βˆ’1
  • Dπ‘Ž=4, 𝑏=βˆ’3

Q10:

Find the values of π‘Ž and 𝑏 that make the function 𝑓 continuous at π‘₯=βˆ’1 and π‘₯=βˆ’6, given that 𝑓(π‘₯)=3π‘₯+11,π‘₯β‰€βˆ’6,π‘Žπ‘₯+𝑏,βˆ’6<π‘₯<βˆ’1,βˆ’5π‘₯+10,π‘₯β‰₯βˆ’1.

  • Aπ‘Ž=125, 𝑏=375
  • Bπ‘Ž=375, 𝑏=125
  • Cπ‘Ž=βˆ’617, 𝑏=βˆ’127
  • Dπ‘Ž=βˆ’127, 𝑏=βˆ’617
  • Eπ‘Ž=125, 𝑏=βˆ’2575

Practice Means Progress

Download the Nagwa Practice app to access 99 additional questions for this lesson!

scan me!

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.