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 lim191𝑥+255.

  • A1019ln
  • Bln19
  • C1
  • D10

Q2:

Determine limln(𝑥1)𝑥2.

Q3:

Determine lim5𝑒5𝑒𝑥.

Q4:

Find lim8𝑒812𝑥+cos.

  • A83
  • B38
  • C38
  • D83

Q5:

Find lim7𝑒7𝑒+1.

  • A565
  • B565
  • C358
  • D358

Q6:

Determine limtan𝑒12𝑥tan.

Q7:

Find lim981749.

  • Alnln97
  • Blnln79
  • C819497lnln
  • D817499lnln

Q8:

Find lim(𝑥+4)𝑥3.

  • A16𝑥3
  • B16𝑥3
  • C4𝑥
  • D4𝑥
  • EThe limit does not exist.

Q9:

Determine lim17131.

  • Aln17
  • Blnln317
  • Cln3
  • Dlnln173

Q10:

Find lim(1+𝑥)1(1+5𝑥)1.

  • A15
  • B35
  • C325
  • D0

Q11:

Find lim(1+𝑥)(1𝑥)(1+𝑥)(1𝑥).

  • A413
  • B134
  • Chas no limit
  • D1

Q12:

Find lim𝑓(𝑥), where 𝑓(𝑥)=𝑥+6𝑥𝑥<2,𝑥+𝑥24𝑥2𝑥>2.ifif

Q13:

Given that lim𝑥(𝑚1)𝑥𝑚𝑥+1=3, determine the value of 𝑚.

Q14:

Determine limsin65𝑥21.

  • A302ln
  • B30
  • C62ln
  • D30𝑒log

Q15:

Determine lim2644𝑥.

  • A162ln
  • B16
  • C92ln
  • D646ln

Q16:

Determine limsin2𝑒3𝑥22𝑥.

  • A52
  • B4
  • C1
  • D12

Q17:

Find limlog12(2𝑥+1)101.

  • A12𝑒10logln
  • B12𝑒10logln
  • C24𝑒10logln
  • D24𝑒10logln

Q18:

Determine lim8𝑒8𝑒9𝑥27.

  • A893ln
  • B98𝑒
  • C89
  • D89𝑒

Q19:

Determine limsin10𝑒10𝑒3𝑥3𝑥sin.

Q20:

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

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

  • A𝑥 dominates ln𝑥.
  • Bln𝑥 dominates 𝑥.

Q21:

Determine lim5𝑒3𝑒3𝑒4𝑒.

  • A53
  • B0
  • C
  • D2
  • E34

Q22:

Find lim2𝑒53𝑒1.

  • A
  • B5
  • C53
  • D2
  • E23

Q23:

Determine limln𝑥𝑥.

Q24:

Determine limln37𝑥+17𝑥.

Q25:

Consider the function 𝑓(𝑥)=𝑥𝑒.

Determine when 𝑓(𝑥)=0.

  • A𝑥=0,𝑥=2
  • B𝑥=0,𝑥=2
  • C𝑥=0,𝑥=2
  • D𝑥=0,𝑥=2
  • E𝑥=2,𝑥=2

Where on the number line is 𝑓(𝑥)<0?

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

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

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