Worksheet: Existence of Limits

In this worksheet, we will practice determining whether the limit of a function at a certain value exists.

Q1:

Discuss the existence of lim𝑓(𝑥) given 𝑓(𝑥)=15𝑥15𝑥<15,𝑥15𝑥15.ifif

  • AThe limit does not exist because lim𝑓(𝑥) exists, but lim𝑓(𝑥) does not exist.
  • BThe limit does not exist because both lim𝑓(𝑥) and lim𝑓(𝑥) exist, but are not equal.
  • CThe limit exists and equals 210.
  • DThe limit does not exist because lim𝑓(𝑥) exists, but lim𝑓(𝑥) does not exist.
  • EThe limit exists and equals 15.

Q2:

Discuss the existence of lim𝑓(𝑥) given 𝑓(𝑥)=𝑥4𝑥<1,20𝑥>1.ifif

  • AThe limit does not exist because 𝑓(1)𝑓(1).
  • BThe limit does not exist because 𝑓(1) does not exist.
  • CThe limit exists and equals 20.
  • DThe limit exists and equals 20.

Q3:

Discuss the existence of lim𝑓(𝑥) given 𝑓(𝑥)=2622𝑥𝑥<4,𝑥2𝑥4𝑥>4.ifif

  • AThe limit does not exist because limlim𝑓(𝑥)𝑓(𝑥).
  • BThe limit exists and equals 5.
  • CThe limit does not exist because lim𝑓(𝑥) does not exist.
  • DThe limit exists and equals 139.
  • EThe limit does not exist because lim𝑓(𝑥) does not exist.

Q4:

Discuss the existence of lim𝑓(𝑥) given 𝑓(𝑥)=𝑥|𝑥+7|3.

  • AThe limit does not exist because lim𝑓(𝑥) does not exist.
  • BThe limit exists and equals 3.
  • CThe limit exists and equals 9.
  • DThe limit does not exist because limlim𝑓(𝑥)𝑓(𝑥).
  • EThe limit does not exist because lim𝑓(𝑥) does not exist.

Q5:

Given that 𝑓(𝑥)=4+𝑥+3𝑥|𝑥+3|3<𝑥<0,2𝑥+40<𝑥<2,ififfind lim𝑓(𝑥).

Q6:

Discuss the existence of lim𝑓(𝑥) given 𝑓(𝑥)=16𝑥+|𝑥2|2𝑥𝑥<2,𝑥+5𝑥>2.ifif

  • AThe limit exists and equals 33.
  • BThe limit exists and equals 22.
  • CThe limit does not exist because limlim𝑓(𝑥)𝑓(𝑥).
  • DThe limit does not exist because lim𝑓(𝑥) does not exist.
  • EThe limit does not exist because lim𝑓(𝑥) does not exist.

Q7:

Find lim𝑓(𝑥) given 𝑓(𝑥)=5𝑥+3𝑥<1,2𝑥1<𝑥<5,𝑥+4𝑥>5.ififif

  • A10
  • B28
  • C 2

Q8:

Discuss the existence of lim𝑓(𝑥) given 𝑓(𝑥)=3𝑥𝜋2<𝑥<0,3𝑥+10<𝑥<3,𝑥27𝑥3𝑥>3.cosififif

  • A l i m 𝑓 ( 𝑥 ) does not exist because limlim𝑓(𝑥)𝑓(𝑥).
  • B l i m 𝑓 ( 𝑥 ) exist and equals 3.
  • C l i m 𝑓 ( 𝑥 ) does not exist because lim𝑓(𝑥) exist, but lim𝑓(𝑥) does not exist.
  • D l i m 𝑓 ( 𝑥 ) does not exist because lim𝑓(𝑥) exist, but lim𝑓(𝑥) does not exist.

Q9:

Given that the function 𝑓(𝑥)=𝑥+𝑎𝑥+𝑏𝑥5𝑥+6𝑥<2,6𝑥𝑥>2ifif has a limit when 𝑥=2, determine the values of 𝑎 and 𝑏.

  • A 𝑎 = 1 2 , 𝑏 = 2 8
  • B 𝑎 = 1 6 , 𝑏 = 3 6
  • C 𝑎 = 1 6 , 𝑏 = 2 8
  • D 𝑎 = 8 , 𝑏 = 1 2

Q10:

Determine lim𝑓(𝑥) using the graph.

  • AThe limit does not exist.
  • B4
  • C 7
  • D 5

Q11:

Given that 𝑓(𝑥)=𝑥+22𝑥<3,5𝑥10𝑥152𝑥4𝑥6𝑥>3,ififdetermine lim𝑓(𝑥).

  • A 5 2
  • B 5 2
  • C 5 4
  • D5

Q12:

Given that 𝑓(𝑥)=𝑥+7𝑥𝑥, determine lim𝑓(𝑥).

  • A7
  • B0
  • CThe limit does not exist.
  • D 7

Q13:

Given the function 𝑓(𝑥)=2𝑥|𝑥4|+5, what is true of lim𝑓(𝑥)?

  • AThe limit exists and equals 13.
  • BThe limit does not exist because limlim𝑓(𝑥)𝑓(𝑥).
  • CThe limit exists and equals 3.

Q14:

Determine lim𝑓(𝑥).

Q15:

What can be said of lim𝑓(𝑥) for the function 𝑓(𝑥)=7𝑥|𝑥|+9𝑥<0,4|𝑥|𝑥+5𝑥>0?ifif

  • A l i m 𝑓 ( 𝑥 ) exists and equals 9.
  • B l i m 𝑓 ( 𝑥 ) exists and equals 1.
  • C l i m 𝑓 ( 𝑥 ) does not exist because lim𝑓(𝑥) exists, but lim𝑓(𝑥) does not exist.
  • D l i m 𝑓 ( 𝑥 ) exists and equals 9.
  • E l i m 𝑓 ( 𝑥 ) does not exist because lim𝑓(𝑥) exists, but lim𝑓(𝑥) does not exist.

Q16:

Given that 𝑓(𝑥)=4𝑥+2𝑥<1,𝑥+3𝑥+61<𝑥<5,4𝑥+30𝑥>5,ififif find lim𝑓(𝑥), if it exists.

Q17:

Find the values of 𝑎 and 𝑏 given lim𝑓(𝑥) and lim𝑓(𝑥) exist where 𝑓(𝑥)=7𝑥9𝑥<1,𝑎𝑥+𝑏1<𝑥<5,8𝑥6𝑥>5.ififif

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

Q18:

Suppose that 𝑓(𝑥)=𝑥1𝑥1𝑥<1,𝑥18𝑥+4𝑥+4𝑥1𝑥>1.ifif What can be said about the existence of lim𝑓(𝑥)?

  • AThe limit exists and equals 67.
  • BThe limit exists and equals 85.
  • CThe limit exists and equals 0.
  • DThe limit does not exist because limlim𝑓(𝑥)𝑓(𝑥).

Q19:

Discuss the existence of lim𝑓(𝑥) given 𝑓(𝑥)=|𝑥2|+32<𝑥<3,𝑥+6𝑥27𝑥3𝑥3<𝑥<9.ifif

  • A l i m 𝑓 ( 𝑥 ) exists and equals 4.
  • B l i m 𝑓 ( 𝑥 ) does not exist because limlim𝑓(𝑥)𝑓(𝑥).
  • C l i m 𝑓 ( 𝑥 ) does not exist because lim𝑓(𝑥) exist, but lim𝑓(𝑥) does not exist.
  • D l i m 𝑓 ( 𝑥 ) does not exist because lim𝑓(𝑥) exist, but lim𝑓(𝑥) does not exist.

Q20:

The figure shows the graph of the function 𝑓(𝑥)=2𝜋𝑥𝑥>0,0.5𝑥0.sin

What is the 𝑥-coordinate of point 𝐴? Give your answer as a fraction.

  • A 1 3 1 4
  • B 1 2 1 3
  • C 1 2 2 5
  • D 1 4 1 5
  • E 1 4 1 5

What is the 𝑥-coordinate of point 𝐵? Give your answer as a fraction.

  • A 1 2 2 7
  • B 1 3 2 5
  • C 1 2 2 5
  • D 1 3 2 7
  • E 1 2 1 3

Setting the 𝑥-coordinate of 𝐴 above as 𝑎, give a formula of the sequence 𝑎, 𝑛1, with 𝐵=(𝑎,0.5).

  • A 𝑎 = 1 2 1 2 𝑛 + 1
  • B 𝑎 = 1 4 4 3 2 5 𝑛 + 4 4 4 3 2 5
  • C 𝑎 = 1 4 4 3 2 5 𝑛 + 1 2 1 3
  • D 𝑎 = 1 2 1 2 𝑛 + 1
  • E 𝑎 = 1 2 1 2 𝑛 + 1 3

What does evaluation of 𝑓(𝑎) with 𝑎 that was previously obtained tell you about a possible value for lim𝑓(𝑥)?

What does evaluation of 𝑓(𝑎) with 𝑎=1𝑛 tell you about a possible value for lim𝑓(𝑥)?

What, therefore, can we conclude about lim𝑓(𝑥)?

  • AWe cannot conclude anything.
  • B l i m 𝑓 ( 𝑥 ) = 1
  • CWe conclude that lim𝑓(𝑥)=0.5.
  • D l i m 𝑓 ( 𝑥 ) = 1
  • EWe conclude that the one-sided limit does not exist.

What law of limits is used to prove that lim𝑓(𝑥)=0.5?

  • AThe limit of the quotient of functions is the quotient of the limits of the given functions if they exist and the denominator is nonzero.
  • BThe limit of a difference of functions is the difference of the limits of the given functions if they exist.
  • CThe limit of the product of functions is the product of the limits of the given functions if they exist.
  • DThe limit of a sum of functions is the sum of the limits of the summand functions if they exist.
  • EThe limit of a constant function as 𝑥 approaches any point is the constant itself.

What can be said about lim𝑓(𝑥)?

  • AIt does not exist because lim𝑓(𝑥) does not exist.
  • BIt exists for some points on the number line and does not for other points.
  • CIt exists and is sometimes 0.5 and sometimes 0.
  • DIt exists and is equal to 0.5 because this is lim𝑓(𝑥).
  • EIt exists and is any value that you like.

Q21:

Discuss the existence of lim[(𝑓+𝑔)(𝑥)], where 𝑓(𝑥)=𝑥+8𝑥<1,𝑥6𝑥1ifif and 𝑔(𝑥)=6𝑥+𝑥𝑥<1,9𝑥𝑥1.ifif

  • A l i m [ ( 𝑓 + 𝑔 ) ( 𝑥 ) ] exists and equals 9.
  • B l i m [ ( 𝑓 + 𝑔 ) ( 𝑥 ) ] exists and equals 5.
  • C l i m [ ( 𝑓 + 𝑔 ) ( 𝑥 ) ] does not exist because lim[(𝑓+𝑔)(𝑥)] exists, but lim[(𝑓+𝑔)(𝑥)] does not exist.
  • D l i m [ ( 𝑓 + 𝑔 ) ( 𝑥 ) ] does not exist because lim[(𝑓+𝑔)(𝑥)] exists, but lim[(𝑓+𝑔)(𝑥)] does not exist.
  • E l i m [ ( 𝑓 + 𝑔 ) ( 𝑥 ) ] exists and equals 4.

Q22:

Determine lim𝑓(𝑥).

  • AThe limit does not exist.
  • B10
  • C3
  • D0

Q23:

Discuss the existence of lim𝑓(𝑥) given 𝑓(𝑥)=202𝑥+𝑥|𝑥|6<𝑥<2,5(𝑥2)𝑥+222<𝑥<14.ifif

  • AThe limit does not exist becauselimlim𝑓(𝑥)𝑓(𝑥).
  • BThe limit exists and equals 24.
  • CThe limit exists and equals 5.
  • DThe limit exists and equals 20.
  • EThe limit exists and equals 0.

Q24:

Given that 𝑓(𝑥)=8𝑥12𝑥+5𝑥3𝑥<12,4𝑥1𝑥>12,ififdetermine lim𝑓(𝑥).

  • A1
  • BThe limit does not exist.
  • C0
  • D 6 7

Q25:

Discuss the existence of lim𝑓(𝑥) given 𝑓(𝑥)=9𝑥|6𝑥3|6𝑥+3𝑥<12,|3𝑥+2|𝑥>12.ifif

  • AThe limit exists and equals 72.
  • BThe limit exists and equals 72.
  • CThe limit exists and equals 112.
  • DThe limit exists and equals 112.
  • EThe limit does not exist because limlim𝑓(𝑥)𝑓(𝑥).

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