Worksheet: L’Hôpital's Rule

In this worksheet, we will practice applying L’Hôpital's rule to evaluate the limits of the indeterminate forms 0/0 and ∞/∞.

Q1:

Find l i m 1 9 1 𝑥 + 2 5 5 .

  • A 1 0 1 9 l n
  • B10
  • C l n 1 9
  • D1

Q2:

Determine l i m l n ( 𝑥 1 ) 𝑥 2 .

Q3:

Determine l i m 5 𝑒 5 𝑒 𝑥 .

Q4:

Find l i m 8 𝑒 8 1 2 𝑥 + c o s .

  • A 3 8
  • B 8 3
  • C 3 8
  • D 8 3

Q5:

Find l i m 7 𝑒 7 𝑒 + 1 .

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

Q6:

Determine l i m t a n 𝑒 1 2 𝑥 t a n .

Q7:

Find l i m 9 8 1 7 4 9 .

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

Q8:

Find l i m ( 𝑥 + 4 ) 𝑥 3 .

  • A 4 𝑥
  • B 4 𝑥
  • CThe limit does not exist.
  • D 1 6 𝑥 3
  • E 1 6 𝑥 3

Q9:

Determine l i m 1 7 1 3 1 .

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

Q10:

Find l i m ( 1 + 𝑥 ) 1 ( 1 + 5 𝑥 ) 1 .

  • A 3 2 5
  • B 1 5
  • C 3 5
  • D0

Q11:

Find l i m ( 1 + 𝑥 ) ( 1 𝑥 ) ( 1 + 𝑥 ) ( 1 𝑥 ) .

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

Q12:

Find l i m 𝑓 ( 𝑥 ) , where 𝑓 ( 𝑥 ) = 𝑥 + 6 𝑥 𝑥 < 2 , 𝑥 + 𝑥 2 4 𝑥 2 𝑥 > 2 . i f i f

Q13:

Given that l i m 𝑥 ( 𝑚 1 ) 𝑥 𝑚 𝑥 + 1 = 3 , determine the value of 𝑚 .

Q14:

Determine l i m s i n 6 5 𝑥 2 1 .

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

Q15:

Determine l i m 2 6 4 4 𝑥 .

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

Q16:

Determine l i m s i n 2 𝑒 3 𝑥 2 2 𝑥 .

  • A 1 2
  • B 5 2
  • C 4
  • D 1

Q17:

Find l i m l o g 1 2 ( 2 𝑥 + 1 ) 1 0 1 .

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

Q18:

Determine l i m 8 𝑒 8 𝑒 9 𝑥 2 7 .

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

Q19:

Determine l i m s i n 1 0 𝑒 1 0 𝑒 3 𝑥 3 𝑥 s i n .

Q20:

Given functions 𝑓 and 𝐹 , which are positive for large values of 𝑥 , we say that 𝐹 dominates 𝑓 as 𝑥 if l i m 𝑓 ( 𝑥 ) 𝐹 ( 𝑥 ) = 0 .

Use l’Hôpital's rule to decide which is dominant as 𝑥 : l n 𝑥 or 𝑥 .

  • A l n 𝑥 dominates 𝑥 .
  • B 𝑥 dominates l n 𝑥 .

Q21:

Determine l i m 5 𝑒 3 𝑒 3 𝑒 4 𝑒 .

  • A 5 3
  • B
  • C 3 4
  • D0
  • E 2

Q22:

Find l i m 2 𝑒 5 3 𝑒 1 .

  • A
  • B 2
  • C 2 3
  • D 5 3
  • E5

Q23:

Determine l i m l n 𝑥 𝑥 .

Q24:

Determine l i m l n 3 7 𝑥 + 1 7 𝑥 .

Q25:

Consider the function 𝑓 ( 𝑥 ) = 𝑥 𝑒 .

Determine when 𝑓 ( 𝑥 ) = 0 .

  • A 𝑥 = 0 , 𝑥 = 2
  • B 𝑥 = 2 , 𝑥 = 2
  • C 𝑥 = 0 , 𝑥 = 2
  • D 𝑥 = 0 , 𝑥 = 2
  • E 𝑥 = 0 , 𝑥 = 2

Where on the number line is 𝑓 ( 𝑥 ) < 0 ?

  • A ( , 0 ) ( 2 , )
  • B ( , )
  • C [ 0 , 2 ]
  • D ( 0 , 2 )
  • E ( , ) ( 0 , 2 )

What is l i m 𝑓 ( 𝑥 ) ?

A sketch of 𝑦 = 𝑓 ( 𝑥 ) on the interval [ 2 , ) is a curve below the 𝑥 -axis that is zero at 𝑥 = 2 and tends to zero as 𝑥 . Knowing that 𝑓 is differentiable, what would an extended Rolle’s theorem tell us about 𝑓 on the interval ( 2 , ) ?

  • AThe function is decreasing before some point 𝑎 ( 2 , ) then starts increasing after that point.
  • BThe function has an inflection point at some point 𝑎 ( 2 , ) , where 𝑓 ( 𝑎 ) = 0 .
  • CThe function has a local minimum at some point 𝑎 ( 2 , ) .
  • DWe cannot get any information about 𝑓 .
  • EThe function has a sharp corner at some point 𝑎 ( 2 , ) .

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