Worksheet: Indeterminate Forms and 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^∞, 1^∞, 0^0, and ∞ - ∞.

Q1:

Use l’HΓ΄pital's rule to determine the limit of π‘›οŽ ο‘ƒ as π‘›β†’βˆž.

Q2:

Given that limο—β†’βˆžπ‘₯=∞ and limο—β†’βˆžοŠ±ο—π‘’=0,by writing π‘₯π‘’οŠ±ο— as π‘₯𝑒, use l’HΓ΄pital's rule to determine limο—β†’βˆžοŠ±ο—π‘₯𝑒.

Q3:

Use l’HΓ΄pital's rule to determine limο—β†’βˆžπ‘₯.οŽ ο‘

Q4:

Use l’HΓ΄pital's rule to determine limsinο—β†’οŠ¦ο€Ό1π‘₯βˆ’1π‘₯.

Q5:

Find limlnο—β†’οŠ¦οŽ©ο€Ό5π‘₯βˆ’83π‘₯.

  • A0
  • B∞
  • Cβˆ’83
  • Dβˆ’βˆž

Q6:

Use L’Hopital’s rule to calculate the value of limsinο—β†’οŠ¦ο—(π‘₯).

Q7:

What is the value of limο—β†’βˆžπ‘₯lnlnοŽ‘οŽ οŽ©ο‘?

  • A2
  • B∞
  • Cβˆ’2
  • D1
  • E0

Q8:

What is the value of limtanο—β†’βˆžο€Όπ‘₯ο€Ό3π‘₯?

Q9:

Let 𝑓(π‘₯)=π‘₯ and 𝑔(π‘₯)=(2π‘₯)ln.

Which function increases faster (has a higher rate of increase) as π‘₯ approaches ∞?

  • A𝑔(π‘₯)
  • B𝑓(π‘₯)

Using that conclusion, what is the value of limο—β†’βˆž(𝑓(π‘₯)βˆ’π‘”(π‘₯))?

  • A∞
  • Bβˆ’βˆž
  • CIt does not exist.
  • D1
  • E0

Q10:

Use L’Hopital’s rule to calculate the value of limο—β†’βˆžο—ο€Ό1βˆ’1π‘₯.

  • A2𝑒
  • Bπ‘’οŠ±οŠ§
  • C𝑒
  • Dβˆ’2𝑒
  • E1

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