Question

To which indeterminate forms does 1 'Hópital's Rule apply directly?

Short answer

Answer: L'Hôpital's Rule can be applied directly to the following indeterminate forms: 1. \(\frac{0}{0}\) 2. \(\frac{\infty}{\infty}\)

Step-by-step solution

Indeterminate Forms:

Indeterminate forms are expressions with an undefined value. There are several commonly known indeterminate forms, such as: 1. \(\frac{0}{0}\) 2. \(\frac{\infty}{\infty}\) 3. \(0 \times \infty\) 4. \(0^0\) 5. \(1^\infty\) 6. \(\infty - \infty\) 7. \(\infty^0\)

L'Hôpital's Rule:

L'Hôpital's Rule is applied to determine the limit of a function that has an indeterminate form of the type \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). According to L'Hôpital's Rule, if: $$\lim_{x \to a} \frac{f(x)}{g(x)}=\frac{0}{0} \ \text{or} \ \frac{\infty}{\infty}$$ And \(f'(x)\) and \(g'(x)\) (the derivatives of \(f(x)\) and \(g(x)\) with respect to \(x\)) are continuous in an interval that contains \(a\), then: $$\lim_{x \to a} \frac{f(x)}{g(x)}=\lim_{x \to a} \frac{f'(x)}{g'(x)}$$ Provided that the limit on the right-hand side exists or is either \(+\infty\) or \(-\infty\).

L'Hôpital's Rule and Indeterminate Forms:

Based on the above explanation, L'Hôpital's Rule can be applied directly to the following indeterminate forms: 1. \(\frac{0}{0}\) 2. \(\frac{\infty}{\infty}\)

Key concepts

Indeterminate Forms

Understanding indeterminate forms is crucial in higher-level calculus, especially when we come across expressions that don't resolve to a clear, defined value. Consider scenarios where, during the evaluation of a limit, you're left with strange expressions like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), and others that seemingly make no sense. Indeterminate forms stir confusion as they don't point to a specific value but rather a 'conflict' in the limit process.

• \(\frac{0}{0}\) - Where both the numerator and denominator approach zero.
• \(\frac{\infty}{\infty}\) - Both parts of the fraction stretch indefinitely.
• \(0 \times \infty\) - Zero times anything should be zero, but infinity challenges this idea.
• \(0^0\), \(1^\infty\), \(\infty - \infty\), and \(\infty^0\) - Each of these represent different levels of mathematical conflict between the concepts of nothingness, boundedness, and infinitude.

Dealing with these requires specialized tools like L'Hôpital's Rule, which provides clarity on the outcome of these limits.

Limits of Functions

The limit of a function is a fundamental concept in calculus that describes the behavior of that function as it approaches a specific point. It gives us a way to talk about values that a function approaches, but does not necessarily reach. This can be particularly useful when functions have breaks or holes, or when they approach infinity.

When evaluating limits, it's not uncommon to encounter indeterminate forms, which is when the standard rules of limits don't give a clear answer. In such cases, techniques like L'Hôpital's Rule help us discern the limit by comparing the rates of change (derivatives) of the function's numerator and denominator. By applying L'Hôpital's Rule properly, we can redefine these puzzling expressions to uncover the true direction and value the function tends toward. It's critical, however, to check first that the rule applies: primarily, the function should exhibit an indeterminate form of type \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).

Continuity and Differentiability

For L'Hôpital's Rule to be the key to unlock a limit involving indeterminate forms, certain conditions concerning continuity and differentiability must be met. Continuity of a function at a point means that there's no abrupt change in value - it's smooth sailing as we move through that point. Differentiability, a step further, implies that not only is the function continuous, but its graph also has a defined slope at that point - it can't be erratic or have sharp turns.

Now, L'Hôpital's Rule hinges on differentiability since it uses derivatives to solve for limits. For the rule to work, the derivatives of both the numerator and denominator - \(f'(x)\) and \(g'((x))\) - must exist and be continuous near the point of concern. This ensures that we can safely replace the original indeterminate limit with a limit involving the derivatives. Remember, if these derivatives don't behave, or if they lead to further indeterminate forms without resolution, L'Hôpital's Rule can't be our lifeline.