Day 14

Lhopitals Rule And Optimization

L'Hôpital's Rule: 0/0 and ∞/∞ Forms Other Indeterminate Forms Optimization

L'Hôpital's Rule: 0/0 and ∞/∞ Forms

Direct substitution sometimes gives an indeterminate answer like $\frac{0}{0}$ that carries no real information on its own. L'Hôpital's Rule replaces the whole fraction with the ratio of the derivatives — often collapsing what looked like an impossible limit into an easy one.

Core Theorem

Suppose

f(a)=g(a)=0

(or both

\to \pm\infty

), and

g'(x) \neq 0

near

a

. Then

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

, provided the right-hand limit exists.
This holds for

x \to a

x \to a^+

x \to a^-

x \to \infty

, and

x \to -\infty

alike. If the new limit is STILL

\frac{0}{0}

\frac{\infty}{\infty}

, apply the rule again.

Watch the TikTok ExplanationL'Hôpital's Rule→

Worked Examples

Read these first — the full solution is shown, with the reasoning behind each move.

Worked Example 01Easy

Evaluate

\lim_{x \to 0} \dfrac{3x-\sin x}{x}

How to think about it

First confirm the limit really is

\frac{0}{0}

\frac{\infty}{\infty}

— L'Hôpital is invalid otherwise. Then differentiate the top and bottom *separately* (not as a quotient) and try the limit again.

Key idea: Confirm it's $\frac{0}{0}$ first, then differentiate numerator and denominator separately.

Full solution

1. Step 1: Confirm the Indeterminate Form

x=0

\frac{3(0)-\sin(0)}{0} = \frac{0}{0}

2. Step 2: Apply L'Hôpital's Rule

\lim_{x \to 0} \frac{3-\cos x}{1}

3. Step 3: Evaluate

\frac{3-\cos(0)}{1} = \frac{3-1}{1} = 2

Worked Example 02Medium

Evaluate

\lim_{x \to 0} \dfrac{1-\cos x}{x^2}

How to think about it

Key idea: You may need to apply L'Hôpital's Rule more than once.

Full solution

1. Step 1: Confirm 0/0 and Apply L'Hôpital's Once

\frac{0}{0} \implies \lim_{x \to 0} \frac{\sin x}{2x}

2. Step 2: Still 0/0 — Apply Again

\lim_{x \to 0} \frac{\sin x}{2x}

is again

\frac{0}{0}

, so differentiate once more:

\lim_{x \to 0} \frac{\cos x}{2}

3. Step 3: Evaluate

\frac{\cos(0)}{2} = \frac{1}{2}

Step-by-Step SOP

1
Confirm 0/0 or ∞/∞ by Direct Substitution
Plug in the limiting value first to verify the indeterminate form actually applies.
2
Differentiate Numerator and Denominator Separately
Find $f'(x)$ and $g'(x)$ independently, then form the new ratio $\frac{f'(x)}{g'(x)}$ .
3
Re-evaluate, and Repeat If Needed
Substitute again. If still indeterminate, differentiate again — as many times as it takes.

Practice — Your Turn

Try each one before opening the hint or the solution.

Practice 01Medium

Evaluate

\lim_{x \to \infty} \dfrac{e^x}{x^2}

Need a hint?

This is

\frac{\infty}{\infty}

— you'll need L'Hôpital's Rule twice before the exponential 'wins.'

Show solution

1. Step 1: Confirm ∞/∞ and Apply L'Hôpital's

\lim_{x \to \infty} \frac{e^x}{2x}

— still

\frac{\infty}{\infty}

2. Step 2: Apply Again

\lim_{x \to \infty} \frac{e^x}{2}

3. Step 3: Conclude

x \to \infty

e^x \to \infty

, so the limit is

\infty

— confirming that exponentials grow faster than any power of

x

Practice 02Easy

Evaluate

\lim_{x \to 1} \dfrac{\ln x}{x-1}

Need a hint?

Confirm the indeterminate form, then differentiate top and bottom.

Show solution

1. Step 1: Confirm 0/0

x=1

\frac{\ln 1}{1-1} = \frac{0}{0}

2. Step 2: Apply L'Hôpital's Rule

\lim_{x \to 1} \frac{1/x}{1} = \lim_{x \to 1} \frac{1}{x}

3. Step 3: Evaluate

\frac{1}{1} = 1

Common Pitfalls

⚠
You MUST Confirm the Indeterminate Form FirstL'Hôpital's Rule only applies to $\frac{0}{0}$ or $\frac{\infty}{\infty}$ — using it on a limit that isn't actually indeterminate gives a wrong answer. Always check by direct substitution first.
⚠
Differentiate Top and Bottom SEPARATELY $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] \neq \frac{f'(x)}{g'(x)}$ — this is NOT the quotient rule. L'Hôpital's Rule differentiates the numerator and denominator independently as two separate functions.
⚠
Re-check After Every ApplicationAfter differentiating once, re-substitute to see if the new limit is still indeterminate. If so, apply L'Hôpital's Rule again — don't stop early.

Other Indeterminate Forms

Products, differences, and exponents can also be indeterminate — $\infty \cdot 0$ , $\infty - \infty$ , and forms like $0^0$ or $1^\infty$ . None of these are ready for L'Hôpital's Rule as written; the trick is always the same: do algebra (or take a logarithm) FIRST to force the expression into a $\frac{0}{0}$ or $\frac{\infty}{\infty}$ shape.

Core Theorem

\infty \cdot 0

: rewrite one factor as a reciprocal in the denominator to create a fraction.

\infty - \infty

: combine into a single fraction first.
Exponential forms (

0^0, \infty^0, 1^\infty

): let

y

equal the expression, take

\ln

of both sides to turn the exponent into a product, evaluate that limit, then exponentiate the result back (

y = e^{\lim \ln y}

Worked Examples

Read these first — the full solution is shown, with the reasoning behind each move.

Worked Example 01Medium

Evaluate

\lim_{x \to 0^+} x\ln x

How to think about it

Products, differences, and exponent forms aren't ready for L'Hôpital as written. Do algebra first (or take

\ln

) to force the expression into a

\frac{0}{0}

\frac{\infty}{\infty}

fraction, then apply the rule.

Key idea: This is $0 \cdot (-\infty)$ — rewrite $x$ as $\frac{1}{1/x}$ to turn it into a fraction.

Full solution

1. Step 1: Rewrite as a Fraction

x\ln x = \dfrac{\ln x}{1/x}

, which is now

\frac{-\infty}{\infty}

x \to 0^+

2. Step 2: Apply L'Hôpital's Rule

\lim_{x \to 0^+} \dfrac{1/x}{-1/x^2} = \lim_{x \to 0^+} \left(-x\right)

3. Step 3: Evaluate

\lim_{x \to 0^+}(-x) = 0

Worked Example 02Hard

Evaluate

\lim_{x \to 0} \left(\dfrac{1}{\sin x} - \dfrac{1}{x}\right)

How to think about it

Key idea: This is $\infty - \infty$ — combine into one fraction over a common denominator first.

Full solution

1. Step 1: Combine into a Single Fraction

\dfrac{1}{\sin x} - \dfrac{1}{x} = \dfrac{x - \sin x}{x \sin x}

, now

\frac{0}{0}

2. Step 2: Apply L'Hôpital's Rule

\lim_{x \to 0} \dfrac{1-\cos x}{\sin x + x\cos x}

— still

\frac{0}{0}

3. Step 3: Apply Again

\lim_{x \to 0} \dfrac{\sin x}{\cos x+\cos x - x\sin x} = \lim_{x \to 0} \dfrac{\sin x}{2\cos x - x\sin x}

4. Step 4: Evaluate

\dfrac{0}{2(1)-0} = \dfrac{0}{2} = 0

Step-by-Step SOP

1
Identify the Indeterminate Form
Determine whether it's $\infty \cdot 0$ , $\infty - \infty$ , or an exponential form ( $0^0$ , $\infty^0$ , $1^\infty$ ).
2
Convert to a Single Fraction (or Take ln First)
Products/differences: combine algebraically into $\frac{0}{0}$ or $\frac{\infty}{\infty}$ . Exponents: set $y$ equal to the expression and take $\ln$ of both sides.
3
Apply L'Hôpital's Rule, Then Exponentiate If Needed
Solve the resulting fraction limit as usual. If you took a log, the final answer is $e$ raised to whatever you found.

Practice — Your Turn

Try each one before opening the hint or the solution.

Practice 01Hard

Evaluate

\lim_{x \to 0^+} x^x

Need a hint?

This is the

0^0

form. Let

y=x^x

, take

\ln

of both sides, and reuse the fact that

\lim_{x \to 0^+} x\ln x = 0

from an earlier example.

Show solution

1. Step 1: Take the Natural Log

Let

y=x^x

. Then

\ln y = x \ln x

2. Step 2: Evaluate the Limit of ln y

From before,

\lim_{x \to 0^+} x\ln x = 0

, so

\lim_{x \to 0^+} \ln y = 0

3. Step 3: Exponentiate Back

Since

\ln y \to 0

y \to e^0 = 1

. So

\lim_{x \to 0^+} x^x = 1

Practice 02AB/BC Standard

Evaluate

\lim_{x \to 0^+} (1+x)^{1/x}

Need a hint?

This is the

1^\infty

form that famously defines the number

e

. Take

\ln

of both sides first.

Show solution

1. Step 1: Take the Natural Log

Let

y=(1+x)^{1/x}

. Then

\ln y = \dfrac{\ln(1+x)}{x}

, which is

\frac{0}{0}

x \to 0

2. Step 2: Apply L'Hôpital's Rule

\lim_{x \to 0} \dfrac{1/(1+x)}{1} = \dfrac{1}{1+0} = 1

3. Step 3: Exponentiate Back

Since

\ln y \to 1

y \to e^1 = e

. So

\lim_{x \to 0^+}(1+x)^{1/x} = e

— this is exactly the limit definition of

e

Common Pitfalls

⚠
Never Apply L'Hôpital's Rule Directly to a Product or Difference $\infty \cdot 0$ and $\infty - \infty$ are NOT ready-made fractions — you must rewrite them as a single fraction (via algebra) before differentiating top and bottom.
⚠
Exponential Forms Always Need a Logarithm FirstThere's no way to directly differentiate an indeterminate exponent like $0^0$ — take $\ln$ of the whole expression, evaluate THAT limit (usually reducing to $\infty \cdot 0$ ), then exponentiate your answer with $e^{(\cdot)}$ at the very end.
⚠
Don't Forget to Exponentiate BackIf you found $\lim \ln y = L$ , the actual answer to the original limit is $e^L$ , not $L$ itself — this last step is easy to forget under time pressure.

Optimization

Optimization problems ask for the best possible outcome — maximum area, minimum material, shortest distance — subject to some constraint. Every single one reduces to the exact same idea: turn a two-variable problem into a one-variable function, then find its absolute extremum.

Core Theorem

Steps for Optimization:
(1) Draw a diagram and label all quantities.
(2) Write the primary equation — the quantity you're optimizing — in terms of two variables.
(3) Write a constraint equation relating those variables, solve for one, and substitute into the primary equation so it depends on only one variable.
(4) Find the absolute extremum of that single-variable function (closed interval method, or First/Second Derivative Test).

Watch the TikTok ExplanationOptimization Problems→

Worked Examples

Read these first — the full solution is shown, with the reasoning behind each move.

Worked Example 01Medium

A farmer has 1200 m of fencing to enclose a rectangular field along a river (no fence is needed on the side facing the river). Find the dimensions that maximize the enclosed area.

How to think about it

Every optimization runs the same way: write the quantity you want to optimize, use the constraint equation to eliminate one variable, then find the absolute extremum of the resulting single-variable function. Check the endpoints of its domain too, and confirm max versus min.

Key idea: Let $x$ be the width of the two perpendicular sides and $y$ the length parallel to the river. The constraint only counts 3 sides of fencing.

Full solution

1. Step 1: Write the Constraint and Primary Equation

Constraint:

2x+y=1200 \implies y=1200-2x

. Primary equation:

A = xy

2. Step 2: Reduce to One Variable

A(x) = x(1200-2x) = 1200x-2x^2

, for

0<x<600

3. Step 3: Find the Critical Point

A'(x) = 1200-4x = 0 \implies x=300

4. Step 4: Confirm the Maximum and State Dimensions

A''(x)=-4<0

, confirming a maximum.

y=1200-600=600

. Maximum area

=300 \times 600 = 180{,}000\ \text{m}^2

Worked Example 02Medium

A 12×12 inch sheet of aluminum has a square of side

x

cut from each corner, and the sides are folded up to form an open-top box. Find

x

that maximizes the volume.

How to think about it

Key idea: The base becomes a $(12-2x)$ by $(12-2x)$ square, and the box height is $x$ .

Full solution

1. Step 1: Write the Volume Function

V(x) = x(12-2x)^2

, for

0<x<6

2. Step 2: Differentiate and Factor

V'(x) = (12-2x)^2 - 4x(12-2x) = (12-2x)\left[(12-2x)-4x\right] = (12-2x)(12-6x)

3. Step 3: Find Critical Points in the Domain

x=6

is an endpoint (excluded), so the only interior critical point is

x=2

4. Step 4: Confirm and Compute

V(0)=V(6)=0

, and

V(2) = 2(8)^2 = 128\ \text{in}^3

— the maximum volume, achieved by cutting

2

-inch squares.

Step-by-Step SOP

1
Draw a Diagram and Label Everything
Identify all the quantities involved and give each one a variable name.
2
Write the Primary and Constraint Equations
Primary: the quantity to optimize, in terms of two variables. Constraint: a relationship between those variables, from the physical setup.
3
Reduce to One Variable and Optimize
Solve the constraint for one variable, substitute into the primary equation, then find its absolute extremum using the derivative tests you already know.

Practice — Your Turn

Try each one before opening the hint or the solution.

Practice 01Hard

A cylindrical can must hold exactly

1000\ \text{cm}^3

. Find the dimensions that minimize the amount of material (surface area) used.

Need a hint?

Solve the volume constraint for

h

in terms of

r

first, then substitute into the surface area formula so it depends only on

r

Show solution

1. Step 1: Write the Constraint and Primary Equation

V = \pi r^2 h = 1000 \implies h = \dfrac{1000}{\pi r^2}

. Surface area:

S = 2\pi r^2 + 2\pi r h

2. Step 2: Reduce to One Variable

S(r) = 2\pi r^2 + \dfrac{2000}{r}

3. Step 3: Differentiate and Solve

S'(r) = 4\pi r - \dfrac{2000}{r^2} = 0 \implies r^3 = \dfrac{500}{\pi} \implies r = \left(\dfrac{500}{\pi}\right)^{1/3} \approx 5.42\ \text{cm}

4. Step 4: Find h and Observe the Classic Result

h = \dfrac{1000}{\pi r^2}

. Substituting

r^3=\frac{500}{\pi}

shows

h=2r

exactly — the optimal can always has height equal to its diameter.

Practice 02Hard

Find the point on the curve

y^2 = 2x

closest to the point

(1,4)

Need a hint?

Minimize the squared distance

D^2=(x-1)^2+(y-4)^2

after substituting

x=\frac{y^2}{2}

so it depends only on

y

Show solution

1. Step 1: Write the Squared Distance in Terms of y

Since

x=\frac{y^2}{2}

D^2(y) = \left(\frac{y^2}{2}-1\right)^2 + (y-4)^2

2. Step 2: Differentiate

\frac{d(D^2)}{dy} = y(y^2-2) + 2(y-4) = y^3-2y+2y-8 = y^3-8

3. Step 3: Solve for the Critical Point

y^3-8=0 \implies y=2

, so

x=\frac{4}{2}=2

4. Step 4: Confirm and State the Answer

\frac{d^2(D^2)}{dy^2}=3y^2>0

y=2

, confirming a minimum. The closest point is

(2,2)

, at distance

\sqrt{(2-1)^2+(2-4)^2}=\sqrt{5}

Practice 03Medium

A company's revenue and cost functions are

R(x)=50x-0.01x^2

and

C(x)=200+20x+0.005x^2

(in dollars, for

x

units). Find the production level that maximizes profit.

Need a hint?

Profit is

P(x)=R(x)-C(x)

— maximize it like any other single-variable function. You should find that the maximum occurs exactly where marginal revenue equals marginal cost.

Show solution

1. Step 1: Write the Profit Function

P(x) = R(x)-C(x) = (50x-0.01x^2)-(200+20x+0.005x^2) = 30x-0.015x^2-200

2. Step 2: Differentiate and Solve

P'(x) = 30-0.03x = 0 \implies x=1000

3. Step 3: Confirm the Maximum

P''(x)=-0.03<0

, confirming a maximum at

x=1000

units — exactly the production level where

R'(x)=C'(x)

(marginal revenue equals marginal cost).

Practice 04Hard

A store sells 200 units of a product per week at 350 dollars each. For every 10-dollar price decrease, weekly sales increase by 20 units. Find the price that maximizes revenue.

Need a hint?

Let

x

be the number of 10-dollar price decrements. Write price and quantity both in terms of

x

, then maximize their product.

Show solution

1. Step 1: Set Up Price and Quantity in Terms of x

Price

=350-10x

. Quantity

=200+20x

2. Step 2: Write the Revenue Function

R(x) = (350-10x)(200+20x) = 70000+5000x-200x^2

3. Step 3: Differentiate and Solve

R'(x) = 5000-400x = 0 \implies x=12.5

4. Step 4: Confirm and State the Answer

R''(x)=-400<0

confirms a maximum. New price

= 350-125 = 225

dollars, and maximum revenue

=R(12.5) = 101{,}250

dollars.

Common Pitfalls

⚠
Always Reduce to a Single Variable Before DifferentiatingThe primary equation almost always starts with two variables — don't differentiate until you've used the constraint to eliminate one of them.
⚠
Don't Skip Confirming Max vs. MinSolving $f'(x)=0$ only gives a candidate — use the Second Derivative Test (or check the domain's endpoints) to confirm it's actually the maximum or minimum you want, not the opposite.
⚠
State the Domain RestrictionsPhysical quantities like side lengths or radii must stay positive — always identify the valid domain (e.g. $0<x<6$ for the box problem) before hunting for extrema, since it can rule out spurious critical points.

Gary Chang

Calculus Educator

5+ Years of Calculus Teaching Experience | AP Calculus Specialist. Dedicated to helping students master calculus through step-by-step logic and clear visualizations.

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Worked Examples

1. Step 1: Confirm the Indeterminate Form

2. Step 2: Apply L'Hôpital's Rule

3. Step 3: Evaluate

1. Step 1: Confirm 0/0 and Apply L'Hôpital's Once

2. Step 2: Still 0/0 — Apply Again

3. Step 3: Evaluate

Confirm 0/0 or ∞/∞ by Direct Substitution

Differentiate Numerator and Denominator Separately

Re-evaluate, and Repeat If Needed

Practice — Your Turn

1. Step 1: Confirm ∞/∞ and Apply L'Hôpital's

2. Step 2: Apply Again

3. Step 3: Conclude

1. Step 1: Confirm 0/0

2. Step 2: Apply L'Hôpital's Rule

3. Step 3: Evaluate

Worked Examples

1. Step 1: Rewrite as a Fraction

2. Step 2: Apply L'Hôpital's Rule

3. Step 3: Evaluate

1. Step 1: Combine into a Single Fraction

2. Step 2: Apply L'Hôpital's Rule

3. Step 3: Apply Again

4. Step 4: Evaluate

Identify the Indeterminate Form

Convert to a Single Fraction (or Take ln First)

Apply L'Hôpital's Rule, Then Exponentiate If Needed

Practice — Your Turn

1. Step 1: Take the Natural Log

2. Step 2: Evaluate the Limit of ln y

3. Step 3: Exponentiate Back

1. Step 1: Take the Natural Log

2. Step 2: Apply L'Hôpital's Rule

3. Step 3: Exponentiate Back

Worked Examples

1. Step 1: Write the Constraint and Primary Equation

2. Step 2: Reduce to One Variable

3. Step 3: Find the Critical Point

4. Step 4: Confirm the Maximum and State Dimensions

1. Step 1: Write the Volume Function

2. Step 2: Differentiate and Factor

3. Step 3: Find Critical Points in the Domain

4. Step 4: Confirm and Compute

Draw a Diagram and Label Everything

Write the Primary and Constraint Equations

Reduce to One Variable and Optimize

Practice — Your Turn

1. Step 1: Write the Constraint and Primary Equation

2. Step 2: Reduce to One Variable

3. Step 3: Differentiate and Solve

4. Step 4: Find h and Observe the Classic Result

1. Step 1: Write the Squared Distance in Terms of y

2. Step 2: Differentiate

3. Step 3: Solve for the Critical Point

4. Step 4: Confirm and State the Answer

1. Step 1: Write the Profit Function

2. Step 2: Differentiate and Solve

3. Step 3: Confirm the Maximum

1. Step 1: Set Up Price and Quantity in Terms of x

2. Step 2: Write the Revenue Function

3. Step 3: Differentiate and Solve

4. Step 4: Confirm and State the Answer

Gary Chang