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

  • AThe function is discontinuous at 𝑥=2 because lim𝑓(𝑥) does not exist.
  • BThe function is continuous at 𝑥=2.
  • CThe function is discontinuous at 𝑥=2 because 𝑓(2)𝑓(𝑥)lim.
  • DThe function is discontinuous at 𝑥=2 because 𝑓(2) is undefined.

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.

  • 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 lim𝑓(𝑥)lim𝑓(𝑥).

Q3:

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

  • AThe function is continuous on .
  • BThe function is discontinuous at 𝑥=1 because 𝑓(1) is undefined.
  • CThe function is continuous at 𝑥=1.
  • DThe function is discontinuous at 𝑥=1 because lim𝑓(𝑥) does not exist.
  • EThe function is discontinuous at 𝑥=1 because 𝑓(1)𝑓(𝑥)lim.

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

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

Q5:

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.

Q6:

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

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

Q7:

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

Q8:

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

  • A12
  • B3
  • C13
  • D2

Q9:

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

  • A653
  • B353
  • C10
  • D53
  • E53

Q10:

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

Q11:

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

  • A3,3
  • B3,3
  • C3
  • D2,2

Q12:

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

  • A12
  • B16
  • C2
  • D16

Q13:

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

  • A2
  • B25
  • C65
  • D6

Q14:

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

  • 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.

Q15:

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

  • A30
  • B198
  • C24
  • D9

Q16:

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

Q17:

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

  • A15
  • B2𝜋51
  • C65
  • D45

Q18:

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

  • A𝑘
  • B0
  • C𝑘
  • D𝑘{0}

Q19:

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

  • AThe function is discontinuous at 𝑥=7 because 𝑓(7)𝑓(𝑥)lim.
  • BThe function is discontinuous at 𝑥=7 because 𝑓(7) is undefined.
  • CThe function is continuous at 𝑥=7.
  • DThe function is continuous at all points in {7}.
  • EThe function is discontinuous at 𝑥=7 because lim𝑓(𝑥) does not exist.

Q20:

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

  • 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).

Q21:

Find the value of 𝑘 which makes the function 𝑓 continuous at 𝑥=0, given 𝑓(𝑥)=2𝑥3𝑥𝑥0,𝑘𝑥=0.sincotifif

  • A6
  • B2
  • C32
  • D23

Q22:

Given 𝑓(𝑥)=6𝑥𝑥<0,6𝑥+1𝑥𝑥𝑥>0.cosifcosif If possible or necessary, define 𝑓(0) so that 𝑓 is continuous at 𝑥=0.

  • AThe function cannot be made continuous at 𝑥=0 because 𝑓(0) is undefined.
  • B𝑓(0)=6 makes 𝑓 continuous at 𝑥=0.
  • CNo value of 𝑓(0) will make 𝑓 continuous because lim𝑓(𝑥) does not exist.
  • DThe function is already continuous at 𝑥=0.

Q23:

Discuss the continuity of the function 𝑓 at 𝑥=0 given 𝑓(𝑥)=5𝑥+7𝑥2𝑥𝑥0,6𝑥=0.sinsinifif

  • 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.

Q24:

Suppose 𝑓(𝑥)=4𝑥+9𝑥3𝜋2,(4𝑥6𝜋)+13𝑥>3𝜋2.sinifif What can be said of the continuity of 𝑓 at 𝑥=3𝜋2?

  • AThe function is continuous at 𝑥=3𝜋2.
  • BThe function is discontinuous at 𝑥=3𝜋2 because 𝑓3𝜋2 is undefined.
  • CThe function is discontinuous at 𝑥=3𝜋2 because lim𝑓(𝑥) does not exist.
  • DThe function is discontinuous at 𝑥=3𝜋2 because 𝑓3𝜋2𝑓(𝑥)lim.

Q25:

Given 𝑓(𝑥)=4𝑥𝑥<0,8𝑥7𝑥𝑥>0.tanifsincosif If possible or necessary, define 𝑓(0) so that 𝑓 is continuous at 𝑥=0.

  • A𝑓(𝑥)=0 would make 𝑓 continuous at 𝑥=0.
  • B𝑓(𝑥)=7 would make 𝑓 continuous at 𝑥=0.
  • CThe function can not be made continuous at 𝑥=0 because lim𝑓(𝑥) does not exist.
  • DThe function is already continuous at 𝑥=0.

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