This image shows a mathematical expression that represents a limit involving indeterminate forms and the application of L'Hôpital's Rule.

Here's a breakdown of the expression:

$\lim_{x \to x_0} \frac{f(x)}{g(x)} = \left(\frac{D}{0}, \frac{\infty}{\infty}\right) = \lim_{x \to x_0} \frac{f'(x)}{g'(x)}$

• On the left: The limit of the ratio $\frac{f(x)}{g(x)}$ as $x \to x_0$. This ratio can lead to indeterminate forms.

• In the middle: The notation $\left( \frac{D}{0}, \frac{\infty}{\infty} \right)$ indicates possible indeterminate forms that occur when evaluating the original limit:

  • $\frac{D}{0}$ represents division by zero, where $D$ is a finite number.
  • $\frac{\infty}{\infty}$ is another indeterminate form.

• On the right: L'Hôpital's Rule is applied when the limit leads to an indeterminate form like $\frac{0}{0}$ or $\frac{\infty}{\infty}$. L'Hôpital's Rule states that if such indeterminate forms occur, the limit of the ratio of the functions can be evaluated as the limit of the ratio of their derivatives:

$\lim_{x \to x_0} \frac{f(x)}{g(x)} = \lim_{x \to x_0} \frac{f'(x)}{g'(x)}$

In summary, the equation in the image is stating that if the limit of $\frac{f(x)}{g(x)}$ as $x \to x_0$ results in an indeterminate form such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then L'Hôpital's Rule can be applied by taking the limit of the ratio of the derivatives $\frac{f'(x)}{g'(x)}$.

Do you want further details or have any specific questions?

Related Questions:

1. What are the conditions for applying L'Hôpital's Rule?
2. How does L'Hôpital's Rule handle other indeterminate forms like $0 \times \infty$?
3. Can L'Hôpital's Rule be applied more than once if the resulting limit is still indeterminate?
4. What are examples of functions that require L'Hôpital’s Rule to evaluate their limits?
5. How does L'Hôpital’s Rule relate to Taylor series expansion for limit evaluation?

Tip:

Always check the form of the limit before applying L'Hôpital's Rule. It must strictly be an indeterminate form like $\frac{0}{0}$ or $\frac{\infty}{\infty}$.