**L’Hospital’s rule** (also spelled *L’Hôpital’s*) is a way to find limits using derivatives when you have indeterminate limits (e.g. {0/0} or {∞/∞}). In those cases, the “usual” ways of finding limits just don’t work. The rule also works for all limits at infinity, or one-sided limits.

L’Hospital’s rule doesn’t work in all cases. For example, it can’t be applied when you have a function of several variables, such as:

That said, if you’re learning about the rule in an early calculus class, it’s unlikely you’ll come across one of these curve balls: you’re much more likely to be given “simple” functions where the rule applies.

## Formal Definition

The formal definition of L’Hospital’s rule looks a little ugly:

However, all it’s really saying is that **in special cases you can use the limit of the derivative to find the limit of the function.** More specifically, the limit of a quotient of functions (e.g. – ∞/∞) equals the quotient of their derivatives. That’s *if *the right-side limit is zero or infinity.

## L’Hospital’s Rule: Example Problem 1

Find

Step 1: **Take the limit of the function** to make sure you have an indeterminate form.

The function is asking you to find the limit at x = 1, so:

- lim ln(x) = ln(1) = 0
- lim (x – 1 ) = 0

The limit on both sides is 0, so it is indeterminate.

Step 2: **Identify f(x) and g(x) from your function** (the two parts in the formal definition of the rule). The top function in the rule is f(x) and the bottom function is g(x). In this case,

- f(x) = ln(x)
- g(x) = (x – 1)

Step 3: **Find the derivative for f(x) and g(x)**:

- D ln(x) = 1/x
- D (x – 1) = 1

Step 3: **Form the new quotient then find the limit**:

## L’Hospital’s Rule: Example Problem 2

Use L’Hospital’s rule to find the limit as x approaches zero for the function ^{sin(x)}⁄_{x} :

Step 1: **Take the limit of the function** to make sure you have an indeterminate form.

lim x→0 ^{sin(x)}⁄_{x} = ^{0}⁄_{0}

If you don’t have an indeterminate form of the limit (i.e. if the numerator and the denominator in the fraction aren’t both zero or infinity), you don’t need L’Hospital’s rule.

Step 2: **Identify f(x) and g(x) from your function**. The top function is f(x) and the bottom function is g(x). In this case, f(x) is sin(x) and g(x) is x.

Step 3: **Find the derivative for f(x) and g(x)**:

f'(sin x) = cos x

f'(x) = 1

Step 4: **Take the limit of the derivative functions** from Step 3:

lim x->0 (cos x) = 1

lim x->0 (x) = 1

Therefore, the limit of the function ^{cos(x)}⁄_{x} = 1/1.

L’Hospital’s rule states that the limit of this derivative function is also the limit of the function itself, so:

lim →0 ^{sin(x)}⁄_{x} = 1

*That’s it!*

## References

Adillon, R. et al. (2015). Mathematics for Economics and Business. Edicions Universitat Barcelona.

Stewart, J. Calculus.