Learning Outcomes

Upon completing this lesson, you will be able to:

• Define indeterminate forms, including $\frac{0}{0}$ and $\frac{\infty}{\infty}$.
• State the conditions under which L'Hôpital's Rule can be applied.
• Apply L'Hôpital's Rule to find the limits of indeterminate forms by differentiating the numerator and denominator.
• Recognize other indeterminate forms $(0 \cdot \infty)$, $(\infty - \infty)$, $(1^\infty)$, $(0^0)$, $(\infty^0)$ and use algebraic manipulation to convert them into a form suitable for L'Hôpital's Rule.

Topics Covered

• Indeterminate Forms
• Statement of L'Hôpital's Rule
• Handling Other Indeterminate Forms

I. Indeterminate Forms

In our study of limits, we have seen that direct substitution is the easiest way to evaluate a limit. However, this method fails when it results in a mathematically ambiguous expression. These expressions are called indeterminate forms. They are "indeterminate" because their value cannot be determined from the form alone.

The two primary indeterminate forms are:

1. $\frac{0}{0}$: This form represents a competition between a numerator that is shrinking to zero and a denominator that is also shrinking to zero. The final value of the limit depends entirely on the relative rates at which they approach zero.
2. $\frac{\infty}{\infty}$: This form represents a competition between a numerator that is growing without bound and a denominator that is also growing without bound. The limit's value depends on which function grows faster.

Encountering an indeterminate form does not mean the limit does not exist. It is a signal that we must use a more powerful technique to analyze the behavior of the functions involved.

II. Statement of L'Hôpital's Rule

L'Hôpital's Rule (pronounced "Lo-pee-tal's") provides a powerful and direct method for resolving the indeterminate forms $\frac{0}{0}$ and $\frac{\infty}{\infty}$. The rule states that if the limit of a quotient of two functions is indeterminate, we can often find the limit by taking the derivative of the numerator and the derivative of the denominator separately.

A. L'Hôpital's Rule Statement

Suppose that $f$ and $g$ are differentiable functions and that $g'(x) \neq 0$ near $c$ (except possibly at $c$). If the limit $\lim_{x \to c} \frac{f(x)}{g(x)}$ produces an indeterminate form of type $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then:

$$ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} $$

Provided that the limit on the right side exists or is $\infty$ or $-\infty$. This rule also applies to limits as $x \to \infty$ or $x \to -\infty$.

Important: This is not the Quotient Rule. We differentiate the numerator and denominator independently of each other.

Example 1

Evaluate the limit:

$$ \lim_{x \to 0} \frac{\sin(x)}{x} $$

Check the form: Direct substitution gives $\frac{\sin(0)}{0} = \frac{0}{0}$. This is indeterminate.

Apply L'Hôpital's Rule: We differentiate the numerator and denominator separately.