# Lecture 15: L'Hopital's Rule

## Calculus I — Differential Calculus

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## Learning Objectives

By the end of this lecture, students will be able to:

1. Identify indeterminate forms (0/0, infinity/infinity, and others)
2. State and apply L'Hopital's Rule
3. Convert other indeterminate forms to 0/0 or infinity/infinity
4. Apply L'Hopital's Rule repeatedly when needed
5. Recognize when L'Hopital's Rule does not apply or is not helpful

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## Lecture Content

### I. Indeterminate Forms

An **indeterminate form** arises when direct substitution in a limit yields an expression whose value cannot be determined without further analysis. The two **primary indeterminate forms** are **0/0** (as in lim_{x -> 0} (sin x)/x) and **infinity/infinity** (as in lim_{x -> infinity} (ln x)/x). L'Hopital's Rule applies directly to both of these.

There are also five **other indeterminate forms** that must be converted before the rule can be used. The form **0 * infinity** can be rewritten as 0/0 or infinity/infinity through algebraic manipulation. The form **infinity - infinity** is handled by combining terms into a single fraction. The forms **0^0, 1^infinity, and infinity^0** are resolved by taking the logarithm, which converts the expression to a product form.

In contrast, **non-indeterminate forms** can be evaluated directly: k/0 (with k nonzero) indicates a limit of plus or minus infinity or that the limit does not exist; 0/k gives a limit of 0; and k * infinity (with k nonzero) gives plus or minus infinity.

### II. L'Hopital's Rule — Statement

**L'Hopital's Rule** states the following: suppose f and g are differentiable near a (except possibly at a) and g'(x) is nonzero near a. If lim_{x -> a} f(x)/g(x) is of the form 0/0 or infinity/infinity, then lim_{x -> a} f(x)/g(x) = lim_{x -> a} f'(x)/g'(x), provided the limit on the right exists or is plus or minus infinity. The rule also applies to one-sided limits and to limits as x approaches plus or minus infinity.

A critical warning: the rule requires differentiating the numerator and denominator **separately**. This is not the quotient rule -- you do not apply the quotient rule formula. You simply replace f(x) with f'(x) and g(x) with g'(x), then re-evaluate the limit.

### III. Examples of 0/0

The classic example is lim_{x -> 0} (sin x)/x. This is a 0/0 form, and applying L'Hopital's Rule gives lim_{x -> 0} cos x / 1 = 1. For lim_{x -> 1} (x^2 - 1)/(x - 1), another 0/0 form, the rule gives lim_{x -> 1} 2x/1 = 2 (though factoring would also work here).

Sometimes L'Hopital's Rule must be applied more than once. For lim_{x -> 0} (e^x - 1 - x)/x^2, the first application gives lim_{x -> 0} (e^x - 1)/(2x), which is still 0/0. Applying the rule again yields lim_{x -> 0} e^x / 2 = 1/2.

<image>A flowchart for applying L'Hopital's Rule. Start: "Evaluate the limit by direct substitution." If the result is 0/0 or infinity/infinity, proceed to "Differentiate numerator and denominator separately." Then "Evaluate the new limit by direct substitution." If the result is still indeterminate, loop back and apply L'Hopital again. If the result is a number, +/- infinity, or DNE, that is the answer. A side box warns: "If the form is NOT indeterminate, STOP — L'Hopital does not apply." Title: "L'Hopital's Rule Decision Flowchart."</image>

### IV. Examples of Infinity/Infinity

For lim_{x -> infinity} (ln x)/x, an infinity/infinity form, L'Hopital's Rule gives lim_{x -> infinity} (1/x)/1 = 0. This confirms that ln x grows much slower than x.

For lim_{x -> infinity} x^2/e^x, another infinity/infinity form, the first application gives lim_{x -> infinity} 2x/e^x, which is still infinity/infinity. Applying the rule again gives lim_{x -> infinity} 2/e^x = 0, confirming that exponentials dominate polynomials. This illustrates the general **growth rate hierarchy** (from slowest to fastest): ln x grows much slower than x^a (for any a > 0), which grows much slower than e^x, which in turn grows much slower than e^{x^2}, and so on.

### V. Converting Other Indeterminate Forms

For the **0 * infinity** form, rewrite f * g as f/(1/g) to obtain 0/0, or as g/(1/f) to obtain infinity/infinity. For example, lim_{x -> 0^+} x ln x = lim_{x -> 0^+} ln x / (1/x), which is now -infinity/infinity. Applying L'Hopital's Rule gives lim_{x -> 0^+} (1/x)/(-1/x^2) = lim_{x -> 0^+} (-x) = 0.

For the **infinity - infinity** form, combine terms into a single fraction. For instance, lim_{x -> 0} (1/x - 1/sin x) = lim_{x -> 0} (sin x - x)/(x sin x), which is now 0/0 and amenable to L'Hopital's Rule.

For **exponential indeterminate forms** (0^0, 1^infinity, infinity^0), let y = f(x)^{g(x)} and take the logarithm: ln y = g(x) * ln f(x). Evaluate the limit of ln y (which is typically a 0 * infinity form), then exponentiate to get the original limit: lim y = e^{lim ln y}. For example, lim_{x -> 0^+} x^x has the form 0^0. Taking logarithms gives ln y = x ln x, which approaches 0 (from the earlier example), so lim x^x = e^0 = 1.

<image>A reference table of the seven indeterminate forms with examples and conversion strategies. Row 1: 0/0 — apply L'Hopital directly. Row 2: infinity/infinity — apply L'Hopital directly. Row 3: 0 * infinity — rewrite as fraction. Row 4: infinity - infinity — combine fractions. Row 5: 0^0 — take logarithm. Row 6: 1^infinity — take logarithm. Row 7: infinity^0 — take logarithm. Each row includes a brief example. Title: "The Seven Indeterminate Forms and How to Handle Them."</image>

### VI. When L'Hopital's Rule Fails or Is Not Helpful

L'Hopital's Rule requires the limit to be indeterminate -- do not apply it to expressions like 0/nonzero or nonzero/0. Sometimes the rule leads to a cycle: for lim_{x -> infinity} (e^x + e^{-x})/(e^x - e^{-x}), applying L'Hopital produces (e^x - e^{-x})/(e^x + e^{-x}), which cycles back to the original form. A better approach in this case is to divide numerator and denominator by e^x.

In other situations, L'Hopital's Rule may produce a more complicated expression rather than simplifying it. When this happens, try algebraic simplification instead. The rule is a powerful tool, but it is not always the most efficient path to a solution.

### VII. Summary of Growth Rates

For limits as x approaches infinity, the following growth rate comparisons are essential. Logarithmic functions grow slower than any positive power of x. Polynomial functions grow slower than any exponential function. Among polynomials, higher degree dominates. Among exponentials, larger base dominates. Formally, lim_{x -> infinity} (ln x)^n / x^p = 0 for any n > 0 and p > 0, and lim_{x -> infinity} x^n / e^{px} = 0 for any n > 0 and p > 0.