Lesson Worksheet: Existence of Limits Mathematics • Higher Education

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

Q1:

Find lim→οŠͺ𝑓(π‘₯) if 𝑓(π‘₯)=5+π‘₯+3π‘₯|π‘₯+3|βˆ’4<π‘₯<0,5π‘₯+40<π‘₯<4.ifif

  • A9
  • B24
  • C1
  • DThe limit does not exist.

Q2:

Discuss the existence of lim→οŠͺ𝑓(π‘₯) given 𝑓(π‘₯)=6π‘₯<4,2π‘₯>4.ifif

  • AThe limit exists and equals 2.
  • BThe limit does not exist because 𝑓(4)≠𝑓(4).
  • CThe limit exists and equals 6.
  • DThe limit does not exist because 𝑓(4) does not exist.

Q3:

Discuss the existence of limο—β†’οŠ±οŠ§π‘“(π‘₯) given 𝑓(π‘₯)=π‘₯βˆ’4,π‘₯<βˆ’1,20,π‘₯>βˆ’1.

  • AThe limit does not exist because 𝑓(βˆ’1)≠𝑓(βˆ’1).
  • BThe limit does not exist because 𝑓(βˆ’1) does not exist.
  • CThe limit exists and equals 20.
  • DThe limit exists and equals βˆ’20.

Q4:

Discuss the existence of limο—β†’οŠ­π‘“(π‘₯) given 𝑓(π‘₯)=13π‘₯+71<π‘₯<7,14π‘₯+77≀π‘₯<8.ifif

  • AThe limit exists and equals 21.
  • BThe limit exists and equals 20.
  • CThe limit does not exist because limlimο—β†’οŠ­ο—β†’οŠ­οŽͺοŽ©π‘“(π‘₯)≠𝑓(π‘₯).
  • DThe limit does not exist because limο—β†’οŠ­οŽͺ𝑓(π‘₯) does not exist.
  • EThe limit does not exist because limο—β†’οŠ­οŽ©π‘“(π‘₯) does not exist.

Q5:

Discuss the existence of limο—β†’οŠ©π‘“(π‘₯) given 𝑓(π‘₯)=|π‘₯βˆ’2|+3,βˆ’2<π‘₯<3,π‘₯+6π‘₯βˆ’27π‘₯βˆ’3π‘₯,3<π‘₯<9.

  • Alimο—β†’οŠ©π‘“(π‘₯) exists and equals 4.
  • Blimο—β†’οŠ©π‘“(π‘₯) does not exist because limlimο—β†’οŠ©ο—β†’οŠ©οŽͺοŽ©π‘“(π‘₯)≠𝑓(π‘₯).
  • Climο—β†’οŠ©π‘“(π‘₯) does not exist because limο—β†’οŠ©οŽͺ𝑓(π‘₯) exist, but limο—β†’οŠ©οŽ©π‘“(π‘₯) does not exist.
  • Dlimο—β†’οŠ©π‘“(π‘₯) does not exist because limο—β†’οŠ©οŽ©π‘“(π‘₯) exist, but limο—β†’οŠ©οŽͺ𝑓(π‘₯) does not exist.

Q6:

Given that 𝑓(π‘₯)=1+π‘₯+3π‘₯|π‘₯+3|βˆ’3<π‘₯<0,4π‘₯+20<π‘₯<5,ifif find limο—β†’οŠ±οŠ©π‘“(π‘₯).

  • A5
  • Bβˆ’14
  • Cβˆ’3
  • DThe limit does not exist.

Q7:

Find limο—β†’οŠ±οŠ―π‘“(π‘₯), where 𝑓(π‘₯)=ο­βˆ’8+|π‘₯+9|,π‘₯β‰ βˆ’9,βˆ’7,π‘₯=βˆ’9.

Q8:

Investigate the behavior of 𝑓(π‘₯)=2ο€Ό1π‘₯cos as π‘₯ tends to 0.

Complete the table of values 𝑓(π‘₯) for values of π‘₯ that get closer to zero.

π‘₯199πœ‹1100πœ‹1999πœ‹11,000πœ‹19,999πœ‹110,000πœ‹
𝑓(π‘₯)β‹―β‹―β‹―β‹―β‹―β‹―
  • Aβˆ’2, 2, βˆ’2, 2, βˆ’2, 2
  • B1, 1, 1, 1, 1, 1
  • Cβˆ’1, 1, βˆ’1, 1, βˆ’1, 1
  • D2, 2, 2, 2, 2, 2

What does this suggest about the graph of 𝑓 close to zero?

  • AThat it oscillates rapidly between βˆ’2 and 2
  • BThat it decreases without bound
  • CThat it changes randomly
  • DThat it approaches 2
  • EThat it increases without bound

Hence, evaluate limο—β†’οŠ¦π‘“(π‘₯).

  • AThe limit does not exist.
  • Bβˆ’2
  • Cβˆ’βˆž
  • D2
  • E∞

Q9:

Find limο—β†’βˆšοŠ«π‘“(π‘₯) if the function 𝑓(π‘₯)=⎧βŽͺ⎨βŽͺ⎩72π‘₯π‘₯<√5,π‘₯βˆ’125√5π‘₯βˆ’5π‘₯>√5.ifif

  • A507√5
  • Bdoes not exist
  • C1752√5
  • D8752√5

Q10:

Discuss the existence of limο—β†’οŠ¨1|π‘₯βˆ’2|.

  • AThe limit does not exist, but limο—β†’οŠ¨1|π‘₯βˆ’2|=βˆ’βˆž.
  • BThe limit exists and is equal to 0.
  • CThe limit does not exist, but limο—β†’οŠ¨1|π‘₯βˆ’2|=∞.
  • DThe limit exists and is equal to 2.
  • EThe limit exists and is equal to 12.

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