Lesson 08: L'Hôpital's Rule & the Mean Value Theorem

What You'll Learn

• Recognize the indeterminate forms 0/0 and ±∞/±∞, apply L'Hôpital's Rule correctly, and identify limits where the rule does not apply because the form is already determinate.
• State the hypotheses of the Mean Value Theorem (MVT), verify they hold, and find the value of c the theorem guarantees.
• Use Rolle's Theorem as the f(a) = f(b) special case of the MVT, cite the Extreme Value Theorem for existence of extrema, and apply the consequences of the MVT (constant and monotonicity results).
• Distinguish the MVT (a statement about a slope/derivative) from the Intermediate Value Theorem (a statement about a value), and reason through CANNOT-style statements.

Content

L'Hôpital's Rule

When you substitute into a quotient and get a meaningless symbol like 0/0 or ∞/∞ — an indeterminate form whose value is not determined by the form alone — you can often resolve the limit by differentiating the top and bottom separately.

L'Hôpital's Rule. If lim f(x)/g(x) gives the form 0/0 or ±∞/±∞, and f and g are differentiable near a with g'(x) ≠ 0 near a, then lim f(x)/g(x) = lim f'(x)/g'(x) provided the right-hand limit exists (or is ±∞). This also holds for one-sided limits and for a = ±∞.

Two conditions govern every use:

1. The form must be 0/0 or ±∞/±∞. If substitution gives 2/0, 0/5, or 3/7, the rule does not apply — the answer is already determined (infinite/DNE, 0, or 3/7 respectively).
2. You differentiate f and g independently. This is not the quotient rule.

Worked (0/0). lim_{x→0} sin(3x)/x substitutes to 0/0. Apply L'Hôpital:

lim_{x→0} sin(3x)/x = lim_{x→0} 3cos(3x)/1 = 3cos(0) = 3

Worked (∞/∞). lim_{x→∞} (ln x)/x is ∞/∞. Apply L'Hôpital:

lim_{x→∞} (ln x)/x = lim_{x→∞} (1/x)/1 = 0

Apply it twice when needed. lim_{x→0} (1 − cos x)/x² is 0/0; differentiating gives (sin x)/(2x), still 0/0; once more gives (cos x)/2 → 1/2. Re-check the form after every application and stop as soon as it becomes determinate.

A disguised derivative. A limit like lim_{x→π} (ln x − ln π)/(x − π) is exactly the definition of the derivative of ln x at x = π, so its value is 1/π. L'Hôpital gives the same result: (1/x)/1 → 1/π.

The Mean Value Theorem

The MVT links the average rate of change of a function over an interval to its instantaneous rate somewhere inside.

Mean Value Theorem. If f is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) with f'(c) = [f(b) − f(a)] / (b − a)

Geometrically: somewhere strictly between a and b, the tangent line is parallel to the secant line through the endpoints.

[GRAPH: increasing curve y = f(x) on [a, b]
- secant line through endpoints (a, f(a)) and (b, f(b))
- tangent line at interior point (c, f(c)) parallel to the secant
- label: secant slope = [f(b) − f(a)]/(b − a) = f'(c)]

Note the deliberate asymmetry: continuity is required on the closed interval [a, b], but differentiability only on the open interval (a, b). The guaranteed c always lives strictly inside, so any solution equal to a or b is discarded.

Finding c — worked. For f(x) = x² on [1, 5]: f is a polynomial, so the hypotheses hold. Secant slope = (25 − 1)/(5 − 1) = 6. With f'(x) = 2x, solve 2c = 6 ⇒ c = 3, which lies in (1, 5).

Rolle's Theorem and the Extreme Value Theorem

Rolle's Theorem. If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there is a c in (a, b) with f'(c) = 0.

Rolle's is just the MVT when the endpoints have equal height: the secant slope is 0, so the guaranteed tangent is horizontal.

Extreme Value Theorem (EVT). If f is continuous on a closed interval [a, b], then f attains an absolute maximum and an absolute minimum on [a, b].

The EVT guarantees extrema exist; it never says where (they may sit at an endpoint).

Consequences of the MVT

• If f'(x) = 0 for every x in an interval, then f is constant there. (Two antiderivatives of the same function differ only by a constant.)
• If f'(x) > 0 throughout, f is increasing; if f'(x) < 0 throughout, f is decreasing.

MVT vs. IVT

One-line hook: IVT is about heights; MVT is about slopes. These existence theorems are a frequent source of CANNOT-style items — statements that contradict a guaranteed conclusion.

Key Takeaways

• Always substitute first; apply L'Hôpital only after confirming a genuine 0/0 or ±∞/±∞ form, and differentiate top and bottom separately, not as a quotient.
• For equal-degree rational functions as x → ±∞, the limit is the ratio of leading coefficients.
• The MVT needs continuity on the closed interval and differentiability on the open interval; the guaranteed c is strictly interior.
• Rolle's Theorem is the f(a) = f(b) special case giving f'(c) = 0; the EVT guarantees a max and min exist for a continuous function on a closed interval.
• Match the conclusion to the question: a guaranteed value is IVT, a guaranteed slope/rate is MVT.

Practice Questions