MTH6131 Chap.12 L'Hôpital's Rule and Indeterminate Forms
L'Hôpital's Rule and Indeterminate Forms
When substitution stops working
A limit can be found by substitution whenever the function is continuous at the point. The notes then show where that fails: sin x over x at zero becomes zero over zero, which is called an indeterminate form because the form alone does not decide the limit.
Besides zero over zero and infinity over infinity, the notes list five more: zero times infinity, infinity minus infinity, one to the power infinity, zero to the power zero, and infinity to the power zero.
The rule and its conditions
Suppose f and g both have derivatives throughout some open interval around the point, and the derivative of g is never zero there.
If both tend to zero, or both tend to plus or minus infinity, then the limit of f/g equals the limit of f′/g′, provided the second limit exists or is infinite. The rule also covers limits at infinity and one-sided limits.
It differentiates the numerator and the denominator separately, and it may be applied again as long as the form is still indeterminate.
Converting the other forms
The rule accepts only quotients. A product of the form zero times infinity is rewritten by moving one factor into the denominator as its reciprocal. A difference of the form infinity minus infinity is combined over a common denominator.
The three power forms use the device from the notes: set y equal to the expression, take the natural logarithm, find the limit of ln y, and exponentiate at the end.
Where the rule misleads
The most expensive error is applying the rule to a quotient that is not indeterminate, which produces a clean-looking wrong answer. The second is differentiating the quotient as a whole instead of its two parts.
A subtler point concerns sin x over x itself: the rule computes it correctly, but it cannot prove it, because the derivative of sine was obtained from that same limit.
What this chapter covers
- 01
What makes a form indeterminate
- 02
The seven indeterminate forms
- 03
L'Hôpital's rule and its hypotheses
- 04
Repeating the rule while the form survives
- 05
Rewriting products and differences as quotients
- 06
The logarithm device for power forms
- 07
Growth rates: polynomials against exponentials
- 08
Misapplications and the circular proof
Worked example · free
A cosine limit that needs two passes
- 1Substitute and name the form.
- 2Apply the rule once and name the new form.
- 2Apply it again and evaluate.
Key terms
- Indeterminate Form
- An expression such as zero over zero produced by substitution that does not by itself determine the value of a limit.
- L'Hôpital's Rule
- The rule that a quotient of the form zero over zero or infinity over infinity has the same limit as the quotient of the derivatives, when that limit exists.
- Power Form
- An indeterminate limit of the type one to the infinity, zero to the zero or infinity to the zero, handled by taking logarithms.
- Growth Rate
- How fast a function increases as its input grows, used to compare, for instance, a power of x with an exponential.
L'Hôpital's Rule and Indeterminate Forms FAQ
Can I use L'Hôpital's rule on any limit of a quotient?
No. The quotient must have the form zero over zero or infinity over infinity at the point, and both functions must be differentiable near it. If substitution gives an ordinary number, that number is the limit, and applying the rule would replace it with something wrong. Substitute first, every time.
How do I handle a limit of the form one to the power infinity?
Set y equal to the expression and take the natural logarithm, which turns the power into a product of the form infinity times zero. Rewrite that as a quotient, apply the rule to find the limit of ln y, and then exponentiate. The limit of (1 + 3/x) to the power x as x grows is e cubed by exactly this route.
Why can the rule not prove that sin x over x tends to one?
Because the rule needs the derivative of sine, and that derivative was found from the limit of sin x over x in the first place. Using the rule to prove the limit would assume what it is meant to show. The rule gives the right number, but a proof uses the geometric squeeze argument.
Exam move
Make a table of the seven forms with one example each, and for each example write the conversion step before any differentiation. Then practise deliberately on quotients that are not indeterminate, so that substituting first becomes a reflex rather than something remembered after a wrong answer.
Working through L'Hôpital's Rule and Indeterminate Forms in MTH6131? Sia is AskSia’s AI Mathematics tutor — ask any MTH6131 L'Hôpital's Rule and Indeterminate Forms question and get a clear, step-by-step explanation grounded in how MTH6131 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
