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Unit 4: Contextual Applications of Differentiation

L'Hôpital's Rule

Using local linearization and L'Hôpital's Rule to approximate values and evaluate indeterminate limits.

Advanced25 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

  • Related Rates

Approximating with a Tangent Line, and Resolving Stubborn Limits

Definition — Local Linearization

L(x) = f(a) + f'(a)(x − a) approximates f(x) for x near a, using the tangent line at a. The approximation is exact at a and grows less accurate farther away; a concave-down function's linearization overestimates nearby, and a concave-up function's underestimates.
L(x)=f(a)+f′(a)(x−a)L(x) = f(a) + f'(a)(x - a)

Worked Example — Approximating a Cube Root

Approximate ∛26 using the linearization of f(x) = x^(1/3) at a = 27. f(27) = 3. f'(x) = (1/3)x^(−2/3), f'(27) = (1/3)(1/9) = 1/27. L(26) = 3 + (1/27)(−1) = 3 − 1/27 ≈ 2.9630. The actual value, ∛26 ≈ 2.9625, is close and slightly less — consistent with f being concave down (f''(x) < 0 for x > 0), so its linearization overestimates.

Some limits resist both direct substitution and algebraic rewriting. If limₓ→c f(x)/g(x) produces 0/0 or ∞/∞, L'Hôpital's Rule replaces it with limₓ→c f'(x)/g'(x), reapplied if the new limit is still indeterminate.

lim⁡x→cf(x)g(x)=lim⁡x→cf′(x)g′(x)(if 00 or ∞∞)\lim_{x \to c}\frac{f(x)}{g(x)} = \lim_{x \to c}\frac{f'(x)}{g'(x)} \quad (\text{if } \tfrac{0}{0} \text{ or } \tfrac{\infty}{\infty})

Worked Example — Resolving a 0/0 Limit

Evaluate limₓ→0 (eˣ − 1)/x. Direct substitution gives (1−1)/0 = 0/0. Differentiate numerator and denominator: eˣ and 1. limₓ→0 eˣ/1 = e⁰ = 1. Check numerically: at x = 0.001, (e^0.001 − 1)/0.001 ≈ 1.0005 — matching.

Derivative Explorer

Point of tangency
1

f(x), with tangent line at x = a

Derivative graph — linked to the graph above; panning or zooming either moves both

What's happening at x = a
  1. At x = 1, f(x) ≈ -2 — the point (1, -2).
  2. The derivative there is f'(1) ≈ 0: the slope of the tangent line, and the instantaneous rate of change of f at this exact point.
  3. That slope is positive, so f is increasing at this point.
  4. The second derivative is positive here, so f is concave up (curving upward) near this point.
  5. Near x = 1, the tangent line y ≈ -2 + 0·(x − 1) is f's best straight-line approximation — the core idea a derivative captures.
Numerical derivative comparison
Forward, backward, and central difference estimates of f'(a) as the step size h shrinks — all three should converge toward the same value.
hForwardBackwardCentral
14-21
0.10.31-0.290.01
0.010.0301-0.02990.0001
0.0010.003-0.0030
Analysis of f(x)
y-intercept
(0, 0)
x-intercepts
(-1.73, 0), (0, 0), (1.73, 0)
Extrema
local max at (-1, 2); local min at (1, -2)
Inflection points
(0, 0)
Vertical asymptotes
none found in view
Horizontal asymptotes
none found
Domain
all real numbers in view
Range (estimated)
approximately [-970, 970]
Analysis of f'(x) — its roots are f's critical points
y-intercept
(0, -3)
x-intercepts
(-1, 0), (1, 0)
Extrema
local min at (0, -3)
Inflection points
none found in view
Vertical asymptotes
none found in view
Horizontal asymptotes
none found
Domain
all real numbers in view
Range (estimated)
approximately [-3, 297]

Tip

Confirm a limit is genuinely 0/0 or ∞/∞ by direct substitution before differentiating — applying L'Hôpital's Rule elsewhere produces a confidently wrong answer.

Common Mistakes

  • Applying L'Hôpital's Rule to a limit that isn't actually 0/0 or ∞/∞.

    limₓ→0 (x+1)/x gives 1/0, not 0/0 — the rule doesn't apply, and using it anyway gives a wrong answer for a limit that doesn't even exist.

  • Using degrees instead of radians when linearizing a trig function.

    Calculus derivative formulas for trig functions assume radians — a Δx measured in degrees has to be converted first.

Key Takeaways

  • Linearization approximates f near a with its tangent line; the error direction follows from concavity.
  • L'Hôpital's Rule replaces an indeterminate limit with the limit of separately-differentiated numerator and denominator.
  • The rule applies only to genuine 0/0 or ∞/∞ forms, confirmed by direct substitution first.

Summary

This closes Unit 4. Unit 5 turns back to a function's own shape, using derivatives to fully classify extrema and concavity.

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