Indeterminate Forms and l’Hôpital’s Rule

Definition Let $f$ and $g$ be defined near $a$, except possibly at $a$, with $g(x) \neq 0$ there. The limit

is of indeterminate type $\frac{0}{0}$ if $f(x) \to 0$ and $g(x) \to 0$ as $x \to a$. It is of type $\frac{\infty}{\infty}$ if $\lvert f(x) \rvert \to \infty$ and $\lvert g(x) \rvert \to \infty$ as $x \to a$. Indeterminate means that the type alone does not determine the limit, which may be any real number, $\pm\infty$, or may not exist.

Definition This is l’Hôpital’s rule. Let $I$ be an open interval containing $a$. Suppose $f$ and $g$ are differentiable on $I$ except possibly at $a$, and $g'(x) \neq 0$ for every $x$ in $I$ with $x \neq a$. Suppose the limit of $f(x)/g(x)$ as $x \to a$ is of type $\frac{0}{0}$ or $\frac{\infty}{\infty}$. If

where $L$ is a real number, $\infty$ or $-\infty$, then also

In the $\frac{\infty}{\infty}$ case it is in fact enough that $\lvert g(x) \rvert \to \infty$, whatever $f$ does.

Definition The rule holds with the same conclusion for one-sided limits $x \to a^+$ and $x \to a^-$, using an interval $(a, a + \delta)$ or $(a - \delta, a)$ in place of $I$, and for $x \to \infty$ or $x \to -\infty$, using an interval $(N, \infty)$ or $(-\infty, N)$. The limit $L$ may be infinite in every case.

Definition If $\lim_{x \to a} f(x) = 0$ and $\lim_{x \to a} \lvert g(x) \rvert = \infty$, the limit of $f(x)g(x)$ is of indeterminate type $0 \cdot \infty$. Rewrite it as a quotient,

which is of type $\frac{0}{0}$ or $\frac{\infty}{\infty}$. The first form needs $g(x) \neq 0$ near $a$, which holds because $\lvert g(x) \rvert \to \infty$. The second needs $f(x) \neq 0$ for $x$ near $a$ with $x \neq a$.

Definition If $\lim_{x \to a} f(x) = \infty$ and $\lim_{x \to a} g(x) = \infty$, the limit of $f(x) - g(x)$ is of indeterminate type $\infty - \infty$. Rewrite the difference as a single quotient, by a common denominator, by rationalizing, or by factoring, and then evaluate that quotient.

Definition A limit of $f(x)^{g(x)}$ with $f(x) > 0$ near $a$ is indeterminate when it is of type $0^0$ ($f \to 0^+$, $g \to 0$), $\infty^0$ ($f \to \infty$, $g \to 0$) or $1^\infty$ ($f \to 1$, $\lvert g \rvert \to \infty$). Write

Because the exponential function is continuous, if $g(x)\ln f(x) \to M$ for a real number $M$, then

If instead $g(x)\ln f(x) \to \infty$, the power tends to $\infty$, and if it tends to $-\infty$, the power tends to $0$.

Definition If $f'/g'$ is again of type $\frac{0}{0}$ or $\frac{\infty}{\infty}$ and $f'$ and $g'$ satisfy the hypotheses, the rule may be applied to $f'/g'$, and so on. The hypotheses must be checked again at each step, and you must stop as soon as the quotient is no longer indeterminate.

Definition If $f'(x)/g'(x)$ has no limit as $x \to a$, finite or infinite, then l’Hôpital’s rule says nothing about the limit of $f(x)/g(x)$, which may still exist. The rule can also go in circles, with each step giving back an equivalent form. In both cases use another method, such as algebra, bounds or the squeeze theorem.