When a limit produces an expression of the form zero multiplied by infinity, direct substitution fails — the result is neither zero nor infinity, and standard arithmetic rules do not apply. This article explains what makes 
indeterminate, how to convert it into a ratio suitable for L'Hôpital's Rule, and how to work through a range of function types where this form arises.
L'Hôpital's Rule and the systematic treatment of indeterminate forms are not on the standard A-Level Mathematics specification (AQA 7357, Edexcel 9MA0, OCR H240, WJEC A420).
This topic is relevant to: A-Level Further Mathematics students seeking enrichment beyond the specification; students preparing for undergraduate Mathematics, Physics, or Engineering; STEP, MAT, and TMUA admissions test candidates encountering limit problems.
Learning Objectives
By the end of this article you should be able to:
- Recognise when a limit has the indeterminate form and explain why direct substitution fails;
- Rewrite a product as a ratio in either
or 
form, choosing the more efficient option; - Apply L'Hôpital's Rule correctly, including applying it more than once where necessary;
- Evaluate limits involving polynomial, trigonometric, exponential, and logarithmic functions in form.
Why Is Indeterminate?
Consider two sequences: one approaching zero, one growing without bound. Their product could converge to any finite value, or diverge to infinity, or oscillate — depending entirely on the rate at which each factor moves. For example:
but 
and 
In all three cases one factor tends to zero and the other to infinity, yet the limits are 1, 
, and 0 respectively. The form therefore carries no inherent value — it is indeterminate.
This is fundamentally different from the definite cases: 
(a fixed finite number multiplied by zero is zero) and 
(a fixed positive number multiplied by infinity is infinity). It is only when both factors are simultaneously tending to their extreme values that the result is unresolvable without further analysis.
L'Hôpital's Rule
L'Hôpital's Rule applies directly to limits in ratio form. If 
and 
are differentiable near 
, and 
near (except possibly at itself), and if the limit 
has the indeterminate form 
or 
, then:
provided the right-hand limit exists (or is 
).
The rule may be applied repeatedly if the ratio remains indeterminate after the first application. It is essential, however, to verify that the indeterminate form condition holds before each application — applying L'Hôpital's Rule when the form is not indeterminate will give a wrong answer.
Common mistake — applying the rule prematurely. L'Hôpital's Rule requires the form or at the point of evaluation.
A form such as 
or 
is not indeterminate — those limits are 0 and 0, respectively and can be read directly. Differentiating the numerator and denominator in a non-indeterminate case produces an incorrect result.
Rewriting as a Ratio
L'Hôpital's Rule cannot be applied directly to a product. The key preliminary step is to rewrite the product 
as one of the following ratios:
The first form places the factor tending to infinity in the numerator and inverts the zero factor into the denominator, producing .
The second form does the reverse, producing . Either form is valid in principle, but one choice typically leads to a cleaner derivative.
As a working rule: if one factor is a trigonometric or logarithmic function, it is usually easier to leave it in the numerator and invert the power or exponential factor.
Worked Example Problems
Example 1 — Trigonometric factor
Evaluate 
As 
, we have 
and 
, so the product has the indeterminate form 
. M1 — identifying the form
Rewrite as a ratio by inverting 
: 
As , both numerator and denominator tend to 0, giving the form. The substitution 
shows this is equivalent to 
, which is the standard sinc limit.
Apply L'Hôpital's Rule — differentiate numerator and denominator separately with respect to 
:
Numerator: 
Denominator: 
M1 — correct application of chain rule on both
The ratio of derivatives is: 
The factors 
cancel exactly, leaving only the cosine.
As , 
, so 
. A1
Note: this result is consistent with the small-angle approximation 
as 
, since 
. L'Hôpital's Rule confirms this rigorously.
Example 2 — Tangent factor
Evaluate 
As , 
and 
, giving the form. M1
Rewrite as by inverting the 
factor: 
Both numerator and denominator tend to 0 as , confirming the 
form.
Differentiate numerator and denominator:
Numerator: 
Denominator: 
M1 — chain rule on sec² form
Form the ratio of derivatives: 
The 
terms cancel; the factor of 3 from the denominator moves to the numerator.
As , 
, so the limit is 
. A1
Again consistent with the small-angle approximation 
: the limit is effectively 
.
Example 3 — Exponential decay
Evaluate 
As , and 
, giving the form. M1
Rewrite 
to place the product in ratio form: 
As both numerator and denominator tend to infinity, giving the 
form. This is typically the cleaner choice here because 
— exponentials differentiate to themselves, simplifying the algebra.
Apply L'Hôpital's Rule: 
M1 — recognising form and differentiating correctly
The form 
is no longer indeterminate — as the denominator grows without bound while the numerator is fixed. A1 
This result encodes the important fact that exponential growth dominates polynomial growth: 
grows far faster than , so their product eventually collapses to zero despite itself growing.
Example 4 — Logarithm near zero
Evaluate 
As 
, 
and 
, giving the 
form. M1
Rewrite as a ratio by writing 
: 
As , the numerator and the denominator 
, giving the 
form. (Note: placing 
in the numerator is preferred over inverting it, since 
is clean, while 
in the denominator position would give a less tractable form.)
Apply L'Hôpital's Rule:
Numerator: 
Denominator: 
M1
Form the ratio: 
Dividing 
by is equivalent to multiplying by 
, which simplifies to 
.A1
This limit is important in probability and information theory (it underlies the convention 
in entropy calculations) and appears frequently in integration by parts problems involving .
When converting to a ratio, you have two options. Invert the zero factor when it is a simple power of (e.g. 
), as this typically produces cleaner derivatives. Invert the infinity factor when it is an exponential (e.g. 
), because exponentials differentiate to themselves and the algebra stays compact. Avoid placing in the denominator — its derivative introduces an additional fraction that rarely simplifies matters.
Common Mistakes That Lose Marks
- Writing
or 
. Neither is valid. These equalities hold only when one factor is a fixed constant, not a limit. - Applying L'Hôpital's Rule without first confirming the indeterminate form. Always verify that both numerator and denominator tend to 0 (or both to ) before differentiating.
- Differentiating the whole fraction as a quotient. L'Hôpital's Rule differentiates numerator and denominator separately; the quotient rule is not used here and gives the wrong result.
- Stopping after one application when the form remains indeterminate. Always re-evaluate the form after each application before deciding whether to stop.
- Forgetting the chain rule when differentiating composite functions such as
or 
— missing the derivative of the inner function is a very common error.
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