**Statement of L'Hospital's Rule:**

L'Hospital's Rule provides a method for evaluating limits of indeterminate forms such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). Specifically, it states:

Let \( f(x) \) and \( g(x) \) be functions that are differentiable on an open interval \( I \) containing \( c \) (except possibly at \( c \) itself). Suppose that:

1. \( \lim_{x \to c} f(x) = \lim_{x \to c} g(x) = 0 \) or \( \lim_{x \to c} f(x) = \lim_{x \to c} g(x) = \pm\infty \).
2. \( g'(x) \neq 0 \) for all \( x \) in \( I \) except possibly at \( c \).

If the limit \( \lim_{x \to c} \frac{f'(x)}{g'(x)} \) exists (or is \( \pm\infty \)), then

\[
\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)},
\]

provided the limit on the right-hand side exists.

**Proof of L'Hospital's Rule:**

We will prove the rule for the case where the limit is of the form \(\frac{0}{0}\) and \( \lim_{x \to c} \frac{f'(x)}{g'(x)} = L \).

**Case 1: \( L \) is a finite number**

Let \( L = \lim_{x \to c} \frac{f'(x)}{g'(x)} \).

We want to show that \( \lim_{x \to c} \frac{f(x)}{g(x)} = L \).

Define a new function \( h(x) = \frac{f(x)}{g(x)} \). We consider the limit as \( x \) approaches \( c \).

Using the Mean Value Theorem for derivatives, for \( x \) near \( c \), there exists a point \( \xi \) between \( x \) and \( c \) such that

\[
\frac{f(x) - f(c)}{x - c} = f'(\xi) \quad \text{and} \quad \frac{g(x) - g(c)}{x - c} = g'(\eta),
\]

where \( \xi \) and \( \eta \) lie between \( x \) and \( c \). Since \( f(c) = g(c) = 0 \), this simplifies to:

\[
\frac{f(x)}{g(x)} = \frac{f'(\xi)}{g'(\eta)}.
\]

Taking the limit as \( x \) approaches \( c \), we have:

\[
\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(\xi)}{g'(\eta)}.
\]

Since \( \xi \) and \( \eta \) both approach \( c \) as \( x \) approaches \( c \), and \( \lim_{x \to c} \frac{f'(x)}{g'(x)} = L \), we obtain:

\[
\lim_{x \to c} \frac{f(x)}{g(x)} = L.
\]

**Case 2: \( L = \pm \infty \)**

In this case, if \( \lim_{x \to c} \frac{f'(x)}{g'(x)} = \pm \infty \), the same reasoning as above applies. The mean value theorem still gives:

\[
\lim_{x \to c} \frac{f(x)}{g(x)} = \pm \infty.
\]

Thus, the proof follows for both cases, and we have established L'Hospital's Rule.

---

Would you like more details on any specific part of the proof or perhaps some examples to illustrate it further?

Here are 5 related questions you might find interesting:

1. How does L'Hospital's Rule apply to limits at infinity?
2. What are the limitations or exceptions to using L'Hospital's Rule?
3. Can L'Hospital's Rule be applied multiple times to the same limit?
4. How does L'Hospital's Rule work with trigonometric functions?
5. How can L'Hospital's Rule be used to solve indeterminate forms beyond \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\)?

**Tip:** Always check if the limit can be simplified before applying L'Hospital's Rule. Sometimes, algebraic manipulation or factoring can lead to a quicker solution.

Explore L'Hospital's Rule, a fundamental concept in calculus used to evaluate limits involving indeterminate forms. This page provides a detailed explanation of the rule's statement, proof, and application scenarios with examples.

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