L'Hôpital's rule (HL only)

IB Mathematics: Analysis and Approaches HL· Unit 5: Calculus· 15 min read

1. Conditions and Core Statement★★☆☆☆HL only⏱ 5 min

📘 Definition

L'Hôpital's Rule

For $\ o x e d lim_{x \to a} \frac{f ( x )}{g ( x )}$

If: (1) $lim_{x \to a} f (x) = lim_{x \to a} g (x) = 0$ or both $\pm \infty$ , (2) $f$ and $g$ are differentiable near $a$ , (3) $g^{'} (x) \neq = 0$ near $a$ , (4) $lim_{x \to a} \frac{f ^{'} ( x )}{g ^{'} ( x )}$ exists, then $lim_{x \to a} \frac{f ( x )}{g ( x )} = lim_{x \to a} \frac{f ^{'} ( x )}{g ^{'} ( x )}$ . The rule works for one-sided limits and limits at infinity.

Example:

Applies to $lim_{x \to 0} \frac{s i n x}{x}$ , which is 0/0 indeterminate.

The most critical step is confirming the original limit is indeterminate. If it is not indeterminate, applying the rule will always give an incorrect result.

📐 Worked Example

Verify that L'Hôpital's rule can be applied to $lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$ , then evaluate the limit.

1
Check the indeterminate form by substituting $x = 2$ :
2
$f(2) = 2^2 - 4 = 0, \quad g(2) = 2 - 2 = 0$
3
We have a 0/0 indeterminate form. Both are polynomials, so differentiable everywhere, and $g^{'} (x) = 1 \neq = 0$ for all $x$ . All conditions are satisfied.
4
Differentiate numerator and denominator separately:
5
$f'(x) = 2x, \quad g'(x) = 1$
6
Evaluate the limit of the ratio of derivatives:
7
$\lim_{x \to 2} \frac{2x}{1} = 4$
8
By L'Hôpital's rule, the original limit equals 4.

2. Basic Indeterminate Forms: 0/0 and ∞/∞★★☆☆☆HL only⏱ 5 min

If after one application of L'Hôpital's rule the new ratio is still indeterminate, you can reapply the rule as long as all conditions still hold.

📐 Worked Example

Evaluate $lim_{x \to \infty} \frac{3 x ^{2} - 2 x + 1}{5 x ^{2} + 4}$ .

1
Check the form as $x \to \infty$ : both numerator and denominator approach $+ \infty$ , so we have an ∞/∞ indeterminate form, all conditions are satisfied.
2
Apply L'Hôpital's rule:
3
$\lim_{x \to \infty} \frac{6x - 2}{10x}$
4
This is still ∞/∞ indeterminate, so apply the rule again:
5
$\lim_{x \to \infty} \frac{6}{10} = \frac{3}{5}$
6
The original limit is $\frac{3}{5}$ , matching the result from dividing numerator and denominator by $x^{2}$ .

✓ Quick check

Test your understanding

Which of the following limits can L'Hôpital's rule be applied to immediately?
- $lim_{x \to 0} \frac{x}{x + 1}$
- $lim_{x \to 0} \frac{1 - c o s x}{x ^{2}}$
- $lim_{x \to 1} \frac{x ^{2} + 1}{x + 1}$
Reveal answer
1 —
Correct. $lim_{x \to 0} (1 - cos x) = 0$ and $lim_{x \to 0} x^{2} = 0$ , so this is 0/0 indeterminate.

3. Other Indeterminate Forms: 0×∞ and ∞−∞★★★☆☆HL only⏱ 6 min

L'Hôpital's rule is only stated for ratios of functions, but we can rearrange other indeterminate forms into 0/0 or ∞/∞ to apply the rule.

For $0 \times \infty$ , rewrite the product as a ratio by moving one term to the denominator: $f (x) g (x) = \frac{f ( x )}{1/ g ( x )}$ , which will become 0/0 or ∞/∞. For $\infty - \infty$ , combine terms into a single fraction, usually via a common denominator.

📐 Worked Example

Evaluate $lim_{x \to 0^{+}} x ln x$ .

1
Check the form: as $x \to 0^{+}$ , $x \to 0$ and $ln x \to - \infty$ , so we have a $0 \times (- \infty)$ indeterminate form.
2
Rewrite as a ratio to get ∞/∞:
3
$x \ln x = \frac{\ln x}{1/x}$
4
Now $ln x \to - \infty$ and $1/ x \to + \infty$ , so we have ∞/∞. All conditions are satisfied. Apply L'Hôpital's rule:
5
Differentiate numerator and denominator:
6
$\frac{d}{dx} \ln x = \frac{1}{x}, \quad \frac{d}{dx} \left(\frac{1}{x}\right) = -\frac{1}{x^2}$
7
Simplify the ratio of derivatives:
8
$\lim_{x \to 0^+} \frac{1/x}{-1/x^2} = \lim_{x \to 0^+} (-x) = 0$
9
So $lim_{x \to 0^{+}} x ln x = 0$ .

4. Indeterminate Powers: $0^0, \infty^0, 1^\infty$★★★☆☆HL only⏱ 5 min

For indeterminate powers of the form $lim f (x)^{g (x)}$ , use natural logarithm to convert the power into a product, which can then be rearranged for L'Hôpital's rule. The standard process is:

Let $y = f (x)^{g (x)}$
Take natural logs: $ln y = g (x) ln f (x)$
Evaluate $lim ln y$ using the 0×∞ method
Exponentiate to get $lim y = e^{l i m l n y}$

📐 Worked Example

Evaluate $lim_{x \to 0^{+}} x^{x}$ .

1
Let $y = x^{x}$ , take natural logs of both sides:
2
$\ln y = x \ln x$
3
We already found that $lim_{x \to 0^{+}} x ln x = 0$ from the previous example, so:
4
$\lim_{x \to 0^+} \ln y = 0$
5
Exponentiate both sides to get the original limit:
6
$\lim_{x \to 0^+} y = \lim_{x \to 0^+} e^{\ln y} = e^0 = 1$
7
So $lim_{x \to 0^{+}} x^{x} = 1$ .

5. Common Pitfalls

Wrong move:

Applying L'Hôpital's rule to a non-indeterminate limit, e.g. $lim_{x \to 0} \frac{x}{x + 1}$ .

Why:

The rule only applies when the original limit is 0/0 or ∞/∞. A non-indeterminate limit can be evaluated directly.

Correct move:

Check that both numerator and denominator approach 0 or both approach ±∞ before applying the rule.

Wrong move:

Differentiating the entire fraction using the quotient rule, instead of differentiating numerator and denominator separately.

Why:

L'Hôpital's rule requires the ratio of individual derivatives, not the derivative of the ratio.

Correct move:

Always write $\frac{f ^{'} ( x )}{g ^{'} ( x )}$ , not $(\frac{f}{g})^{'} (x)$ .

Wrong move:

Stopping after one application when the derivative ratio is still indeterminate.

Why:

If the first ratio is still indeterminate, you need to reapply the rule to get the correct result.

Correct move:

Check the form of the derivative ratio after each application, and reapply if it is still indeterminate.

Wrong move:

Applying the rule directly to 0×∞, ∞−∞, or indeterminate powers without rearrangement.

Why:

L'Hôpital's rule is only defined for ratios of functions, so applying it directly to other forms is invalid.

Correct move:

Always rearrange the expression into a 0/0 or ∞/∞ ratio before applying L'Hôpital's rule.

6. Quick Reference Cheatsheet

Indeterminate Form	How to apply L'Hôpital's rule
0/0, ∞/∞	Apply directly after checking conditions
0 × ∞	Rewrite as $\frac{f}{1/ g}$ to get 0/0 or ∞/∞
∞ − ∞	Combine into a single fraction to get 0/0 or ∞/∞
$0^{0}, \infty^{0}, 1^{\infty}$	Take natural log to convert to 0 × ∞

7. Frequently Asked

Do I need to check the conditions before applying L'Hôpital's rule?

Yes, IB exam markers always require you to confirm that the limit is indeterminate and that all other conditions (differentiability, non-zero derivative of denominator) are satisfied before applying the rule. Skipping this step will lose marks.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Evaluate limit of rational function
2022 · 2
Find limit of exponential function
2023 · 1
Evaluate 0×∞ indeterminate limit

What's Next

L'Hôpital's rule is a core HL tool for evaluating indeterminate limits, which are required for improper integrals, testing series convergence, and finding limits of sequences in IB AA HL. It is commonly tested alongside differentiation and series topics, and confirming the rule's conditions will help you avoid losing easy marks in exams. Mastery of this method will make more advanced calculus topics significantly easier to work through.