Using L'Hopital's Rule for Infinity/Infinity Indeterminate Forms

What an Infinity/Infinity Form Means

Some limits produce expressions that look like . This is called an indeterminate form. It does not mean that the answer is automatically infinity, 1, or any other particular number.

The important idea is that both the numerator and denominator are becoming arbitrarily large in magnitude, but we do not yet know which one grows faster. Their relative growth determines the limit.

For example, as , both and grow without bound. But the quotient can still approach a finite number, infinity, zero, or another value depending on how quickly the two functions grow.

What L'Hopital's Rule Says

L'Hopital's Rule gives us a way to compare the rates at which the numerator and denominator are changing. If a limit has the form and the required conditions are satisfied, we can differentiate the numerator and denominator separately:

The key point is that you differentiate the numerator and denominator separately. You do not use the quotient rule.

How to Recognize When to Use It

Before applying L'Hopital's Rule, substitute the limiting value or determine the end behavior. You want to establish that the original expression actually produces the indeterminate form.

Evaluate the limit informally first. Determine what the numerator and denominator approach.
Confirm the form is . Both numerator and denominator must become unbounded in the relevant direction.
Differentiate the numerator.
Differentiate the denominator.
Evaluate the new limit. If it is still an indeterminate form, L'Hopital's Rule may be applied again.
Stop when the limit can be evaluated directly.

Worked Examples

Example 1: A Basic Infinity/Infinity Limit

Evaluate .

Step 1: As , the numerator approaches and the denominator approaches . This is an form.

Step 2: Differentiate the numerator and denominator:

Step 3: Evaluate the resulting limit:

Result:

Example 2: Polynomial Growth Rates

Evaluate .

Step 1: As , both the numerator and denominator approach , so the form is .

Step 2: Apply L'Hopital's Rule:

Step 3: The new limit is still , so apply the rule again:

Step 4: Apply it one more time:

Result:

Example 3: Exponential Growth

Evaluate .

Step 1: As , both and approach infinity. Therefore, the form is .

Step 2: Differentiate numerator and denominator:

Step 3: Evaluate the new limit:

Result:

Example 4: A Limit Requiring L'Hopital's Rule Twice

Evaluate .

Step 1: As , both the numerator and denominator approach infinity. This is .

Step 2: First application of L'Hopital's Rule:

Step 3: The new expression is still , so apply the rule again:

Step 4: Evaluate the resulting limit:

Result:

Why L'Hopital's Rule Works

The intuition is about relative growth rates. If both functions become extremely large, their actual sizes are less important than how quickly they are growing.

L'Hopital's Rule replaces the original functions with their derivatives, allowing us to compare those growth rates directly. For example, exponential functions eventually grow much faster than polynomial functions, which is why a limit such as approaches zero.

Common Mistakes

Using L'Hopital's Rule without checking the form

Do not automatically differentiate numerator and denominator whenever you see a limit. First establish that the expression is actually an indeterminate form such as . L'Hopital's Rule is not a general-purpose limit rule.

Using the quotient rule

L'Hopital's Rule is not the quotient rule. You do not differentiate the entire fraction with the quotient rule. Instead, differentiate the numerator and denominator separately.

Assuming infinity divided by infinity equals 1

Infinity is not an ordinary number. The expression does not automatically equal 1. The answer depends on the relative growth of the numerator and denominator.

Stopping after one application

After differentiating, check the new limit again. If it is still , you may need to apply L'Hopital's Rule again.

L'Hopital's Rule vs. Algebraic Simplification

L'Hopital's Rule is powerful, but it is not always the best first method. If ordinary algebra can simplify the limit immediately, that may be cleaner and faster.

For example, if the expression is a quotient of polynomials, comparing the highest powers can often reveal the answer without repeated differentiation. L'Hopital's Rule becomes especially useful when the functions involve exponentials, logarithms, trigonometric functions, or combinations that are difficult to simplify algebraically.

Practice Problems

Question 1

Evaluate .

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Question 2

Evaluate .

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Question 3

Evaluate .

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Question 4 (Repeated L'Hopital)

Evaluate .

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