AHL 5.13—Limits and L’Hopitals

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 $x\to \pm \infty$).

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:
${\lim }_{x\to a}\frac{f(x)}{g(x)}={\lim }_{x\to a}\frac{f^{\prime }(x)}{g^{\prime }(x)}$
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.