CAL-09

L'Hôpital snapshot

lim f/g = lim f'/g' when 0/0 or ∞/∞ and the latter limit exists.

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LimitsL'Hôpital

Governing equation

\lim\dfrac f g=\lim\dfrac{f'}{g'}

where

f': f' (—)
g': g' (—)
f'/g': L'Hôpital ratio (—)

Lecture brief

Historical brief

Newton and Leibniz (1670s), Taylor, the fundamental theorem and the trapezoid rule are how change became a number. The lab differentiates, integrates and linearises in one variable. This sheet (CAL-09 — L'Hôpital snapshot) is the form associated with L'Hôpital. Working symbols: $f'$ , $g'$ $\rightarrow$ $f'/g'$ . Cauchy's mean-value theorem on a shrinking interval. Hypotheses matter — counterexamples exist.

Purpose

Purpose: compute

f'/g'

from

f'

g'

in Calculus via

\lim\dfrac f g=\lim\dfrac{f'}{g'}

lim f/g = lim f'/g' when 0/0 or ∞/∞ and the latter limit exists. Use it when a real calculus question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

f' = 2.000\,\mathrm{—}

g' = 4.000\,\mathrm{—}

, the governing relation

\lim\dfrac f g=\lim\dfrac{f'}{g'}

yields

f'/g' = 0.5000\,\mathrm{—}

. Enter f' and g' near the point; the sheet reports their ratio. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

f' f'2 —g' g'4 —

Outputs

L'Hôpital ratio f'/g'0.5000 —

Reading speed

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Narration of this film

Enter f' and g' near the point; the sheet reports their ratio.

Cauchy's mean-value theorem on a shrinking interval. Hypotheses matter — counterexamples exist.

Reading speed

Watch on YouTube