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Worksheet: Applying L’Hôpital's Rule for Limits with 0/0 Output

Q1:

Find l i m π‘₯ β†’ 0 π‘₯ 1 9 βˆ’ 1 √ π‘₯ + 2 5 βˆ’ 5 .

  • A l n 1 9
  • B10
  • C1
  • D 1 0 1 9 l n

Q2:

Determine l i m l n π‘₯ β†’ 2 ( π‘₯ βˆ’ 1 ) π‘₯ βˆ’ 2 .

Q3:

Determine l i m π‘₯ β†’ 0 1 9 π‘₯ 1 0 π‘₯ 5 𝑒 βˆ’ 5 𝑒 π‘₯ .

Q4:

Find l i m π‘₯ β†’ 4 π‘₯ 3 πœ‹ 2 πœ‹ 8 8 𝑒 βˆ’ 8 βˆ’ 1 2 π‘₯ + c o s .

  • A 3 8
  • B βˆ’ 8 3
  • C βˆ’ 3 8
  • D 8 3

Q5:

Find l i m π‘₯ β†’ 0 5 π‘₯ 8 π‘₯ 7 𝑒 βˆ’ 7 βˆ’ 𝑒 + 1 .

  • A 3 5 8
  • B βˆ’ 5 6 5
  • C 5 6 5
  • D βˆ’ 3 5 8

Q6:

Determine l i m t a n π‘₯ β†’ 0 2 π‘₯ 𝑒 βˆ’ 1 2 π‘₯ t a n .

Q7:

Find l i m π‘₯ β†’ 2 π‘₯ π‘₯ 9 βˆ’ 8 1 7 βˆ’ 4 9 .

  • A l n l n 9 7
  • B 8 1 7 4 9 9 l n l n
  • C l n l n 7 9
  • D 8 1 9 4 9 7 l n l n

Q8:

Find l i m β„Ž β†’ 0 4 4 ( π‘₯ + 4 β„Ž ) βˆ’ π‘₯ 3 β„Ž .

  • A 4 π‘₯ 4
  • B 1 6 π‘₯ 3 4
  • C 4 π‘₯ 3
  • D 1 6 π‘₯ 3 3
  • EThe limit does not exist.

Q9:

Determine l i m π‘₯ β†’ 0 2 π‘₯ 2 π‘₯ 1 7 βˆ’ 1 3 βˆ’ 1 .

  • A l n 1 7
  • B l n l n 3 1 7
  • C l n 3
  • D l n l n 1 7 3

Q10:

Find .

  • A0
  • B
  • C
  • D

Q11:

Find l i m π‘₯ β†’ 0 1 3 1 3 4 4 ( 1 + π‘₯ ) βˆ’ ( 1 βˆ’ π‘₯ ) ( 1 + π‘₯ ) βˆ’ ( 1 βˆ’ π‘₯ ) .

  • A 4 1 3
  • B1
  • Chas no limit
  • D 1 3 4

Q12:

Suppose What can be said of the continuity of 𝑓 at π‘₯ = πœ‹ ?

  • A The function is discontinuous at π‘₯ = πœ‹ because l i m π‘₯ β†’ πœ‹ 𝑓 ( π‘₯ ) does not exist.
  • B The function is continuous on ℝ .
  • C The function is discontinuous at π‘₯ = πœ‹ because 𝑓 ( πœ‹ ) is undefined.
  • D The function is continuous at π‘₯ = πœ‹ .
  • E The function is discontinuous at π‘₯ = πœ‹ because 𝑓 ( πœ‹ ) β‰  𝑓 ( π‘₯ ) l i m π‘₯ β†’ πœ‹ .

Q13:

Find l i m π‘₯ β†’ 2 𝑓 ( π‘₯ ) , where

Q14:

Given that l i m π‘₯ β†’ βˆ’ 1 2 π‘₯ βˆ’ ( π‘š βˆ’ 1 ) π‘₯ βˆ’ π‘š π‘₯ + 1 = βˆ’ 3 , determine the value of π‘š .

Q15:

Determine l i m s i n π‘₯ β†’ 0 π‘₯ βˆ’ 6 5 π‘₯ 2 βˆ’ 1 .

  • A βˆ’ 3 0
  • B βˆ’ 6 2 l n
  • C βˆ’ 3 0 𝑒 l o g 2
  • D βˆ’ 3 0 2 l n

Q16:

Determine l i m π‘₯ β†’ 0 6 + π‘₯ 2 βˆ’ 6 4 4 π‘₯ .

  • A 9 2 l n
  • B 6 4 6 l n
  • C16
  • D 1 6 2 l n

Q17:

Determine l i m s i n π‘₯ β†’ 0 π‘₯ 2 𝑒 βˆ’ 3 π‘₯ βˆ’ 2 βˆ’ 2 π‘₯ .

  • A βˆ’ 1
  • B βˆ’ 4
  • C βˆ’ 5 2
  • D 1 2

Q18:

Find l i m l o g π‘₯ β†’ 0 π‘₯ βˆ’ 1 2 ( 2 π‘₯ + 1 ) 1 0 βˆ’ 1 .

  • A βˆ’ 2 4 𝑒 1 0 l o g l n
  • B βˆ’ 1 2 𝑒 1 0 l o g l n
  • C βˆ’ 1 2 𝑒 1 0 l o g l n
  • D βˆ’ 2 4 𝑒 1 0 l o g l n

Q19:

Determine l i m π‘₯ β†’ 3 π‘₯ 3 8 𝑒 βˆ’ 8 𝑒 9 π‘₯ βˆ’ 2 7 .

  • A 8 9
  • B 9 8 𝑒 1 3
  • C 8 9 3 l n
  • D 8 9 𝑒 3

Q20:

Determine l i m s i n π‘₯ β†’ 0 3 π‘₯ 3 π‘₯ 1 0 𝑒 βˆ’ 1 0 𝑒 3 π‘₯ βˆ’ 3 π‘₯ s i n .