AHL 5.13—Limits and L’Hopitals: IB DP Math AA Notes (HL)

What Are Indeterminate Forms?

When evaluating limits, substituting the value of $x$ directly sometimes produces an expression that has no clear numerical meaning. These are called indeterminate forms — the limit may still exist, but direct substitution cannot tell us what it is.

Indeterminate Form

An expression obtained when evaluating a limit that is mathematically undefined, meaning the value of the limit cannot be determined from the form alone. Examples include $\frac{0}{0}$ , $\frac{\infty}{\infty}$ , $0 \cdot \infty$ , $\infty - \infty$ , $0^{0}$ , $1^{\infty}$ , and $\infty^{0}$ .

The two indeterminate forms most directly handled by L'Hôpital's Rule are:

$\frac{0}{0}$ : Both numerator and denominator approach zero as $x \to a$ .
$\frac{\infty}{\infty}$ : Both numerator and denominator grow without bound as $x \to a$ (or as $x \to \pm \infty$ ).

Note

Other indeterminate forms like $0 \cdot \infty$ , $\infty - \infty$ , $0^{0}$ , $1^{\infty}$ , and $\infty^{0}$ cannot be directly handed to L'Hôpital's Rule — but many of them can be algebraically rearranged into a $\frac{0}{0}$ or $\frac{\infty}{\infty}$ form first.

Warning

Not every limit that looks tricky is indeterminate. For example, $lim_{x \to 0} \frac{x ^{2} + 1}{x + 2} = \frac{1}{2}$ by direct substitution — no special techniques needed. Always try direct substitution first!

L'Hôpital's Rule

If $f$ and $g$ are differentiable near $x = a$ , and the limit $lim_{x \to a} \frac{f ( x )}{g ( x )}$ produces an indeterminate form of type $\frac{0}{0}$ or $\frac{\infty}{\infty}$ , then:
$lim_{x \to a} \frac{f ( x )}{g ( x )} = lim_{x \to a} \frac{f ^{'} ( x )}{g ^{'} ( x )}$
provided the limit on the right exists (or is $\pm \infty$ ).

In plain terms: if you have a qualifying indeterminate form, differentiate the numerator and denominator separately (not as a quotient using the quotient rule!) and then take the limit again.

Exam Tip

L'Hôpital's Rule applies not just when $x \to a$ for a finite value $a$ , but also when $x \to \infty$ or $x \to - \infty$ . This makes it particularly powerful for analysing long-run behaviour of functions.

Common Mistake

A very common error is applying the quotient rule instead of differentiating the numerator and denominator separately. L'Hôpital's Rule says:
$lim \frac{f}{g} = lim \frac{f ^{'}}{g ^{'}} NOT lim (\frac{f}{g})^{'}$
These are completely different things!

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What Are Indeterminate Forms?

L'Hôpital's Rule

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