Worksheet: Continuity at a Point

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=2, given 𝑓(𝑥)=𝑥+8𝑥4𝑥2,3𝑥=2.ifif

  • A The function is continuous at 𝑥=2.
  • B The function is discontinuous at 𝑥=2 because 𝑓(2) is undefined.
  • C The function is discontinuous at 𝑥=2 because 𝑓(2)𝑓(𝑥)lim.
  • D The function is discontinuous at 𝑥=2 because lim𝑓(𝑥) does not exist.

Q2:

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.

  • A 𝑓 ( 8 ) = 0 would make 𝑓 continuous at 𝑥=8.
  • BThe function is already continuous at 𝑥=8.
  • CThe function cannot be made continuous at 𝑥=8 because lim𝑓(𝑥)lim𝑓(𝑥).
  • D 𝑓 ( 8 ) = 6 would make 𝑓 continuous at 𝑥=8.

Q3:

Suppose 𝑓(𝑥)=𝑥+1𝑥1𝑥<1,62𝑥10𝑥1.ifif What can be said of the continuity of 𝑓 at 𝑥=1?

  • A The function is continuous on .
  • B The function is discontinuous at 𝑥=1 because 𝑓(1)𝑓(𝑥)lim.
  • C The function is discontinuous at 𝑥=1 because lim𝑓(𝑥) does not exist.
  • D The function is discontinuous at 𝑥=1 because 𝑓(1) is undefined.
  • E The function is continuous at 𝑥=1.

Q4:

Find the values of 𝑎 and 𝑏 that make the function 𝑓 continuous at 𝑥=1 and 𝑥=6, given that 𝑓(𝑥)=3𝑥+11𝑥6,𝑎𝑥+𝑏6<𝑥<1,5𝑥+10𝑥1.ififif

  • A 𝑎 = 1 2 7 , 𝑏 = 6 1 7
  • B 𝑎 = 1 2 5 , 𝑏 = 3 7 5
  • C 𝑎 = 6 1 7 , 𝑏 = 1 2 7
  • D 𝑎 = 1 2 5 , 𝑏 = 2 5 7 5
  • E 𝑎 = 3 7 5 , 𝑏 = 1 2 5

Q5:

Given 𝑓(𝑥)=𝑥+𝑥2𝑥1, if possible or necessary, define 𝑓(1) so that 𝑓 is continuous at 𝑥=1.

  • A 𝑓 ( 1 ) = 3 makes 𝑓 continuous at 𝑥=1.
  • BThe function cannot be made continuous at 𝑥=1 because 𝑓(1) is undefined.
  • CThe function is already continuous at 𝑥=1.
  • DNo value of 𝑓(1) will make 𝑓 continuous because lim𝑓(𝑥) does not exist.

Q6:

Given 𝑓(𝑥)=𝑥64𝑥+𝑥20, if possible or necessary, define 𝑓(4) so that 𝑓 is continuous at 𝑥=4.

  • ANo value of 𝑓(4) will make 𝑓 continuous because lim𝑓(𝑥) does not exist.
  • BThe function cannot be made continuous at 𝑥=4 because 𝑓(4) is undefined.
  • CThe function is already continuous at 𝑥=4.
  • D 𝑓 ( 4 ) = 1 6 3 makes 𝑓 continuous at 𝑥=4.

Q7:

Discuss the continuity of the function 𝑓 at 𝑥=5 given 𝑓(𝑥)=8𝑥+1𝑥5,𝑥25𝑥125𝑥>5.ifif

  • A The function is discontinuous at 𝑥=5 because lim𝑓(𝑥)𝑓(5).
  • B The function is continuous at 𝑥=5.
  • C The function is discontinuous at 𝑥=5 because 𝑓(5) is undefined.
  • D The function is discontinuous at 𝑥=5 because lim𝑓(𝑥) does not exist.

Q8:

Find the value of 𝑎 that makes 𝑓 continuous at 𝑥=3, given that 𝑓(𝑥)=𝑥+𝑥(𝑎3)3𝑎𝑥3𝑥3,7𝑥+6𝑥=3.ifif

Q9:

Setting 𝑓(𝑎)=54 and 𝑓(𝑥)=𝑥𝑎𝑥𝑎 when 𝑥𝑎 makes 𝑓 continuous at 𝑥=𝑎. Determine 𝑎.

  • A 1 3
  • B2
  • C3
  • D 1 2

Q10:

Determine the value of 𝑎 that makes 𝑓 continuous at 𝑥=0, given 𝑓(𝑥)=56𝑥57𝑥3𝑥𝑥0,𝑎𝑥=0.sintanifif

  • A 5 3
  • B 5 3
  • C 1 0
  • D 6 5 3
  • E 3 5 3

Q11:

Discuss the continuity of the function 𝑓 at 𝑥=0, given 𝑓(𝑥)=𝑥5𝑥𝑥0,5𝑥=0.sinifif

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

Q12:

The function 𝑓(𝑥)=7|𝑥|𝑥+17𝑥<0,𝑎+9𝑥𝑥0ifcosif is continuous at 𝑥=0. Determine the possible values of 𝑎.

  • A 3 , 3
  • B 3
  • C 2 , 2
  • D 3 , 3

Q13:

Determine the value of 𝑎 that makes the function 𝑓 continuous at 𝑥=𝜋4, given 𝑓(𝑥)=2𝑥+9𝑥4+4𝑥𝑥𝜋4,3𝑎𝑥=𝜋4.sintansinifif

  • A2
  • B 1 6
  • C 1 2
  • D 1 6

Q14:

Find the value of 𝑘 that makes the function 𝑓 continuous at 𝑥=𝜋4, given that 𝑓(𝑥)=62𝑥+4𝑥2𝑥+2𝑥𝜋4,5𝑘𝑥=𝜋4.sintansinifif

  • A 6
  • B 2 5
  • C 6 5
  • D 2

Q15:

Discuss the continuity of the function 𝑓 at 𝑥=𝜋2, given 𝑓(𝑥)=7𝑥+7𝑥𝑥𝜋2,62𝑥1𝑥>𝜋2.sincosifcosif

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

Q16:

Determine the value of 𝑎 that makes the function 𝑓 continuous at 𝑥=0, given that 𝑓(𝑥)=68𝑥8𝑥2𝑥𝑥0,𝑎6𝑥=0.sintanifif

  • A9
  • B30
  • C24
  • D198

Q17:

Find the value of 𝑘 which makes the function 𝑓 continuous at 𝑥=0, given that 𝑓(𝑥)=2𝑥4𝑥7𝑥𝑥0,7𝑘𝑥=0.sintanifif

  • A2
  • B 2 4 9
  • C 8 7
  • D 8 4 9
  • E 1 2

Q18:

Discuss the continuity of the function 𝑓 at 𝑥=𝜋2, given 𝑓(𝑥)=8+7𝑥𝑥<𝜋2,7+5𝑥𝑥𝜋2.cosifsinif

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

Q19:

Let 𝑓(𝑥)=𝜋𝑥5𝑥𝑥<0,𝜋𝑎+6𝜋5𝑥𝑥0.sinifcosif Find all values of 𝑎 that make 𝑓 continuous at 𝑥=0.

  • A 2 𝜋 5 1
  • B 6 5
  • C 4 5
  • D 1 5

Q20:

Let 𝑓(𝑥)=7𝑥+78𝑘𝑥𝑥0,24𝑥𝑥=0.cosifsinif Find all values of 𝑘 that make 𝑓 continuous at 𝑥=0.

  • A 𝑘 { 0 }
  • B 𝑘
  • C 𝑘
  • D0

Q21:

Suppose 𝑓(𝑥)=1𝑥+5𝑥𝑥𝑥0,4+15𝑥𝑥>0.cossinifcosif What can be said of the continuity of 𝑓 at 𝑥=0?

  • A The function is discontinuous at 𝑥=0 because lim𝑓(𝑥) does not exist.
  • B The function is discontinuous at 𝑥=0 because 𝑓(0)𝑓(𝑥)lim.
  • C The function is continuous at 𝑥=0.
  • D The function is discontinuous at 𝑥=0 because 𝑓(0) is undefined.
  • E The function is continuous on .

Q22:

Find the value of 𝑘 which makes the function 𝑓 continuous at 𝑥=4, given 𝑓(𝑥)=(𝑥4)4𝑥16𝑥4,𝑘𝑥=4.sinifif

  • A4
  • B 1 4
  • C 1
  • D 1 4
  • E 4

Q23:

Discuss the continuity of the function 𝑓 at 𝑥=7, given that 𝑓(𝑥)=|𝑥+7|𝑥2,𝑥+3𝑥>2.ifif

  • A The function is continuous at all points in {7}.
  • B The function is discontinuous at 𝑥=7 because 𝑓(7)𝑓(𝑥)lim.
  • C The function is continuous at 𝑥=7.
  • D The function is discontinuous at 𝑥=7 because lim𝑓(𝑥) does not exist.
  • E The function is discontinuous at 𝑥=7 because 𝑓(7) is undefined.

Q24:

Discuss the continuity of the function 𝑓 at 𝑥=0, given that 𝑓(𝑥)=9𝑥|𝑥|+2𝑥0,86|𝑥|𝑥𝑥>0.ifif

  • A The function is continuous at 𝑥=0.
  • B The function is continuous on {0}.
  • C The function is discontinuous at 𝑥=0 because lim𝑓(𝑥) does not exist.
  • D The function is discontinuous at 𝑥=0 because 𝑓(0)𝑓(𝑥)lim.
  • E The function is discontinuous at 𝑥=0 because 𝑓(0) is undefined.

Q25:

Discuss the continuity of the function 𝑓 at 𝑥=0, given that 𝑓(𝑥)=6𝑥𝑥𝑥4𝑥𝑥<0,𝑥+5𝑥+4𝑥0.sintanifif

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

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