L'Hôpital's rule — proof using linear approximation, 0 - 0, ∞ - ∞, other indeterminate forms: Foundations

Before you can use L'Hôpital's rule, you must be fluent in a handful of ideas the parent note quietly assumes. This page builds each one from absolutely nothing, in the order they lean on each other.

1. The symbol $lim$ and the arrow $\to$

Picture it. Imagine walking along the number line toward the point $a$ but never quite stepping on it. At each position $x$ you record the height $h (x)$ . The limit $L$ is the height those recordings settle onto.

Figure — L'Hôpital's rule — proof using linear approximation, 0 - 0, ∞ - ∞, other indeterminate forms

Why the topic needs it. L'Hôpital's rule is a statement about limits — it never asks for the value at $a$ (often that value doesn't even exist). It only cares about the approach. That is why we may have $f (a) = g (a) = 0$ and still ask a meaningful question about $\frac{f}{g}$ near $a$ . See Limits — Definition & Laws for the full machinery.

2. Functions $f$ , $g$ and the ratio $\frac{f ( x )}{g ( x )}$

The parent note studies the ratio $\frac{f ( x )}{g ( x )}$ — one machine's output divided by another's, at the same input $x$ .

Picture it. Two curves drawn on the same axes. At each $x$ you read off two heights, $f (x)$ (top) and $g (x)$ (bottom), and divide top by bottom.

3. The forms $\frac{0}{0}$ and $\frac{\infty}{\infty}$ — "indeterminate"

Why "indeterminate", not "undefined"? $\frac{1}{0}$ is undefined (division by zero, full stop). But $\frac{0}{0}$ is worse and better at once: both numbers are approaching zero, so the pattern is a question — "which shrinks faster?" — that still has a real answer.

$lim_{x \to 0} \frac{s i n x}{x} = 1, lim_{x \to 0} \frac{s i n x}{x ^{2}} = \infty, lim_{x \to 0} \frac{x ^{2}}{s i n x} = 0.$

All three look like $\frac{0}{0}$ yet land on $1$ , $\infty$ , and $0$ . That single fact is the reason the whole rule must exist. The complete gallery lives in Indeterminate Forms.

4. The symbol $\infty$

Picture it. A curve rising up the page and never levelling off. There is no top height — only the behaviour of climbing.

Why the topic needs it. The second direct case of the rule is $\frac{\infty}{\infty}$ , and one endpoint of many problems is $x \to \infty$ (" $x$ marches off to the right forever"). Since $\infty$ is not a real number, we may never "plug it in" — we can only take limits.

5. Slope, and the derivative $f^{'} (x)$

This is the engine of the whole topic, so we build it slowly.

The symbol $Δ$ (Greek "delta") just means "the change in." So $Δ x$ = change in $x$ .

Why measure a curve's slope? A straight line has one slope everywhere. A curve's steepness changes as you move along it. The derivative captures the slope at a single point by shrinking the sideways step to zero:

$f^{'} (x) = lim_{Δ x \to 0} \frac{f ( x + Δ x ) - f ( x )}{Δ x} .$

Why this exact tool, and not another? The indeterminate question was "how fast does each part vanish?" Speed-of-change is literally the definition of the derivative. No other single quantity measures the rate at which a function moves. That is why L'Hôpital reaches for $f^{'}$ and $g^{'}$ and nothing else.

6. The tangent line & linear approximation

Let's earn every piece of that formula:

$f (a)$ — the height where we touch.
$f^{'} (a)$ — the slope at that touch point (from §5).
$(x - a)$ — how far sideways we've moved from the touch point.
$f^{'} (a) (x - a)$ — slope × sideways move = rise. Add it to the starting height.

Why the topic needs it. The parent's whole intuitive proof is: replace top and bottom by their tangent lines, watch the constant terms $f (a), g (a)$ vanish (they're zero!), and cancel the shared $(x - a)$ . What's left is $\frac{f ^{'} ( a )}{g ^{'} ( a )}$ — the ratio of speeds. See Linear Approximation & Tangent Lines.

$\frac{f ( x )}{g ( x )} \approx \frac{f ( a ) = 0 + f ^{'} ( a ) ( x - a )}{= 0 g ( a ) + g ^{'} ( a ) ( x - a )} = \frac{f ^{'} ( a ) ( x - a )}{g ^{'} ( a ) ( x - a )} = \frac{f ^{'} ( a )}{g ^{'} ( a )} .$

7. Continuity (so $f (a) = g (a) = 0$ makes sense)

Why the topic needs it. The proof defines $f (a) = g (a) = 0$ "by continuity." That phrase only works if the functions are continuous — then the value at $a$ equals the limit there, letting us pin both to $0$ cleanly and cancel the constant terms.

8. The Mean Value Theorem — the rigorous backbone

Picture it. Drive from $a$ to $b$ . Your average speed over the trip must have been your actual speed at some instant in between — you can't average $60$ without ever hitting $60$ .

Why the topic needs it. The tangent-line argument secretly assumed $f^{'}, g^{'}$ are continuous. The airtight proof upgrades to the Cauchy MVT, which delivers an exact point $c$ between $a$ and $x$ with $\frac{f ( x )}{g ( x )} = \frac{f ^{'} ( c )}{g ^{'} ( c )}$ — no cancellation hand-waving needed. Full details in Mean Value Theorem.

9. $ln$ , $e^{x}$ — the log/exponent toolkit

Why the topic needs it. Three of the seven indeterminate forms are exponent forms: $0^{0}, \infty^{0}, 1^{\infty}$ . A derivative can't "see" an exponent until $ln$ pulls it down to the ground floor, turning $f^{g}$ into $g ln f$ — a product we can then wrestle into a fraction. This is also why $lim (1 + 1/ x)^{x} = e$ works. Growth-rate comparisons like $ln x$ vs $x$ live in Exponential & Logarithm Growth Rates.

10. Beyond first order — the Taylor idea (preview)

That tower is the Taylor Series, the deep generalisation of everything on this page.

Prerequisite map

Equipment checklist

Cover the right side and test yourself.

What does

lim_{x \to a} h (x) = L

mean in plain words?

x

gets arbitrarily close to

a

(never arriving),

h (x)

settles onto

L

Why is

\frac{0}{0}

called indeterminate, not undefined?

Both parts are shrinking to zero, so the pattern is a question ("which shrinks faster?") that still has a genuine answer.

\infty

a number you can plug in?

No — it means "grows past every bound"; you may only take limits.

What does the derivative

f^{'} (x)

measure geometrically?

The instantaneous slope (steepness / speed of change) of

f

x

Write the tangent-line / linear approximation formula.

L (x) = f (a) + f^{'} (a) (x - a)

Why does the constant term vanish in the

0/0

proof?

Because

f (a) = g (a) = 0

, so the tangent lines reduce to

f^{'} (a) (x - a)

and

g^{'} (a) (x - a)

What does continuity let us assert about

f (a)

That

f (a) = lim_{x \to a} f (x)

, so we can pin it to

0

with no jump.

State the Mean Value Theorem in one sentence.

Somewhere inside the interval, the instantaneous slope equals the average slope.

Why does

ln

appear for

1^{\infty}

and

0^{0}

ln

drags the exponent down (

ln f^{g} = g ln f

), turning an invisible exponent into a product a derivative can handle.

Why is the derivative the right tool for

\frac{0}{0}

The form asks "how fast does each part vanish?" — and speed-of-change is exactly what a derivative is.

Connections

Limits — Definition & Laws — the meaning of $lim$ and $\to$ .
Indeterminate Forms — the full gallery of $\frac{0}{0}$ -type patterns.
Linear Approximation & Tangent Lines — the picture that powers the proof.
Mean Value Theorem — the rigorous backbone (Cauchy MVT).
Exponential & Logarithm Growth Rates — why $ln$ and $e^{x}$ enter.
Taylor Series — the deep generalisation.
L'Hôpital's Rule (parent) — where all this equipment is used.