Day 15

Lhopitals Rule

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

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→

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 Exercises

Example 01Easy

Evaluate

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

NEED A HINT?

Confirm it's

\frac{0}{0}

first, then differentiate numerator and denominator separately.

SHOW DETAILED EXPLANATION

Step 1: Confirm the Indeterminate Form

x=0

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

Step 2: Apply L'Hôpital's Rule

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

Step 3: Evaluate

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

Example 02Medium

Evaluate

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

NEED A HINT?

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

SHOW DETAILED EXPLANATION

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

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

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}

Step 3: Evaluate

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

Example 03Medium

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 DETAILED EXPLANATION

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

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

— still

\frac{\infty}{\infty}

Step 2: Apply Again

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

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

Example 04Easy

Evaluate

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

NEED A HINT?

Confirm the indeterminate form, then differentiate top and bottom.

SHOW DETAILED EXPLANATION

Step 1: Confirm 0/0

x=1

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

Step 2: Apply L'Hôpital's Rule

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

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}

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 Exercises

Example 01Medium

Evaluate

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

NEED A HINT?

This is

0 \cdot (-\infty)

— rewrite

x

\frac{1}{1/x}

to turn it into a fraction.

SHOW DETAILED EXPLANATION

Step 1: Rewrite as a Fraction

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

, which is now

\frac{-\infty}{\infty}

x \to 0^+

Step 2: Apply L'Hôpital's Rule

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

Step 3: Evaluate

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

Example 02Hard

Evaluate

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

NEED A HINT?

This is

\infty - \infty

— combine into one fraction over a common denominator first.

SHOW DETAILED EXPLANATION

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}

Step 2: Apply L'Hôpital's Rule

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

— still

\frac{0}{0}

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}

Step 4: Evaluate

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

Example 03Hard

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 DETAILED EXPLANATION

Step 1: Take the Natural Log

Let

y=x^x

. Then

\ln y = x \ln x

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

Step 3: Exponentiate Back

Since

\ln y \to 0

y \to e^0 = 1

. So

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

Example 04AB/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 DETAILED EXPLANATION

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

Step 2: Apply L'Hôpital's Rule

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

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.

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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Confirm 0/0 or ∞/∞ by Direct Substitution

Differentiate Numerator and Denominator Separately

Re-evaluate, and Repeat If Needed

Practice Exercises

Step 1: Confirm the Indeterminate Form

Step 2: Apply L'Hôpital's Rule

Step 3: Evaluate

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

Step 2: Still 0/0 — Apply Again

Step 3: Evaluate

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

Step 2: Apply Again

Step 3: Conclude

Step 1: Confirm 0/0

Step 2: Apply L'Hôpital's Rule

Step 3: Evaluate

Identify the Indeterminate Form

Convert to a Single Fraction (or Take ln First)

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

Practice Exercises

Step 1: Rewrite as a Fraction

Step 2: Apply L'Hôpital's Rule

Step 3: Evaluate

Step 1: Combine into a Single Fraction

Step 2: Apply L'Hôpital's Rule

Step 3: Apply Again

Step 4: Evaluate

Step 1: Take the Natural Log

Step 2: Evaluate the Limit of ln y

Step 3: Exponentiate Back

Step 1: Take the Natural Log

Step 2: Apply L'Hôpital's Rule

Step 3: Exponentiate Back

Gary Chang