- Topic
- L’Hôpital’s rule
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
L’Hôpital’s rule practice question 1
L’Hôpital’s rule may only be applied to a limit that is genuinely indeterminate. Which of the following limits is of the form or ?
For each limit, what do the top and the bottom tend to on their own? Which pairs of values leave the limit undecided?
Substitute the limiting value into numerator and denominator separately. Only and are indeterminate; and are not.
Theory UBC Math 100 §6CLP-1 §3.7Key results
Worked solution
Strategy
Check the form before reaching for the rule. Three of these limits can be read off directly; only one needs work.
Working
Indeterminate: l’Hôpital applies (and gives e⁰/1 = 1).
None of these is indeterminate, so the rule does not apply; used anyway it can give a wrong answer, as cos x / x shows.
Answer
The answer is the limit of e to the x minus 1 over x, which has the form 0 over 0.
Where this goes wrong
Applying l’Hôpital to cos x / x, which “gives” −sin x / 1 → 0. The form is 1/0, the limit does not exist, and the rule was never allowed.
Leans on
Evaluating limits by substitution, from Limits
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