Question

Explain the steps used to apply l'Hôpital's Rule to a limit of the form \(0 / 0\)

Short answer

Question: Explain the main steps in applying l'Hôpital's Rule to solve a limit of the form \(\frac{0}{0}\). Answer: In order to apply l'Hôpital's Rule to a limit of the form \(\frac{0}{0}\), follow these main steps: 1. Identify the indeterminate form by checking if both the numerator and denominator go to zero as the variable approaches the specified value. 2. Verify that the functions in the numerator and denominator are differentiable, and the limit of their derivatives exists or results in another indeterminate form. 3. Apply l'Hôpital's Rule by differentiating the functions in the numerator and denominator and rewriting the limit using the derivatives. 4. Check if the new limit can be resolved, either finding a definite value or identifying another indeterminate form. 5. Repeat steps 2 to 4 as necessary until a definite limit is found, or it is determined that the limit does not exist.

Step-by-step solution

Identify the indeterminate form

First, make sure that the given limit is an indeterminate form, particularly \(\frac{0}{0}\). To do this, attempt to compute the limit by simply plugging in the specified value of the variable. If both the numerator and denominator go to zero, you've identified an indeterminate form suitable for applying l'Hôpital's Rule.

Verify that l'Hôpital's Rule can be applied

Before applying l'Hôpital's Rule, verify that the functions in the numerator and denominator are both differentiable (i.e., they can be differentiated). Additionally, confirm that the limit of their derivatives exists or results in another indeterminate form. If both conditions are satisfied, l'Hôpital's Rule can be applied.

Apply l'Hôpital's Rule

According to l'Hôpital's Rule, if the limit \(\lim_{x\to a} \frac{f(x)}{g(x)}\) is of the form \(\frac{0}{0}\) or \(\frac{\pm\infty}{\pm\infty}\), then \(\lim_{x\to a} \frac{f(x)}{g(x)} = \lim_{x\to a} \frac{f'(x)}{g'(x)}\). In this step, differentiate both the numerator and the denominator separately with respect to the variable. After finding the derivatives, rewrite the limit using the derivatives of the given functions in the numerator and denominator.

Check if the new limit can be resolved

Now that you have applied l'Hôpital's Rule to find a new limit, calculate or evaluate the new limit. If you can find a definite value for the limit, the process is completed. If the new limit is still an indeterminate form like \(\frac{0}{0}\) or \(\frac{\pm\infty}{\pm\infty}\), then you must repeat steps 2 to 4 to see if the limit can be resolved.

Repeat Steps 2 to 4 if necessary

If the new limit obtained after applying l'Hôpital's Rule is still an indeterminate form, repeat steps 2 to 4 until you reach a definite limit or determine that the limit does not exist. Remember that l'Hôpital's Rule can be applied multiple times as long as the conditions to use it are always satisfied. By following these steps, one can successfully apply l'Hôpital's Rule to a limit of the form \(\frac{0}{0}\).

Key concepts

Indeterminate Forms

When you're working through limits in calculus, you'll often encounter what are known as 'indeterminate forms'. These peculiar mathematical expressions don't automatically have a clear value and can't be directly evaluated without further analysis. Among the most common indeterminate forms is \( \frac{0}{0} \). At first glance, this ratio may seem nonsensical because dividing anything by zero is undefined. However, within the context of limits, this particular form implies that as we approach a certain point, both the numerator and the denominator shrink toward zero, creating an uncertainty about the limit's true value.

Other indeterminate forms include expressions like \( \frac{\infty}{\infty} \) (where both the numerator and denominator grow without bound), \( 0 \cdot \infty \) (where zero is being multiplied by infinity), \( \infty - \infty \) (subtraction of infinities), \( 0^0 \) (zero raised to the power of zero), and \( \infty^0 \) and \( 1^\infty \) (infinity raised to the power of zero and one raised to the power of infinity, respectively). These scenarios don’t yield straightforward results, so specialised techniques like l'Hôpital's Rule are used to clarify what these limits actually are.

• Indeterminate forms require special techniques to resolve.
• \(\frac{0}{0}\) is a common indeterminate form in calculus.
• Recognizing an indeterminate form is the first step before applying methods like l'Hôpital's Rule.

Differentiation

Differentiation is a fundamental operation in calculus that is at the core of many techniques, including l'Hôpital's Rule. It's the process of determining the derivative of a function, which represents the rate at which the function's value changes with respect to a change in its input value. Essentially, when you differentiate a function, you're finding a new function that tells you the slope of the tangent to the original function's curve at any point.

When applying differentiation in l'Hôpital's Rule, we are interested in the derivatives of the numerator and denominator functions separately. The derivative is a powerful tool because, under the right conditions, it can transform indeterminate forms into determinate ones that are possible to evaluate. For example, the indeterminate form \( \frac{0}{0} \) can often be resolved by differentiating the numerator and the denominator and then taking the limit of the resulting fraction.

• Differentiation finds the rate of change of a function.
• Derivatives transform functions and can clarify indeterminate limits.
• The derivative is essential for applying l'Hôpital's Rule.

Limits in Calculus

Limits are central to the study of calculus and play a vital role in the understanding of the continuity and behavior of functions at specific points. A limit attempts to describe the value that a function approaches as the input approaches a particular point, which might not necessarily be the function's actual value at that point. In calculus, limits can be straightforward, but they can also form indeterminate expressions that are not immediately resolvable.

To effectively deal with these tricky situations, mathematicians use strategies like l'Hôpital's Rule. This rule is particularly useful when dealing with the indeterminate form \( \frac{0}{0} \) and other forms like \( \frac{\infty}{\infty} \). It states that if a limit yields an indeterminate form, then under certain conditions, one can find the limit of the ratio of their derivatives instead. It's a highly practical tool that hinges upon the principles of differentiation and gives us a method to find limits that would otherwise be very difficult to determine directly.

• Limits describe the behavior of functions at specific points.
• L'Hôpital's Rule is used to resolve certain kinds of indeterminate limits.
• The rule allows the evaluation of the limit of derivatives instead of the original functions.