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Chapter 4: Problem 2

Why are special methods, such as l'H么pital's Rule, needed to evaluate indeterminate forms (as opposed to substitution)?

Short Answer

Expert verified
In order to evaluate indeterminate forms like \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0^\infty\), \(\infty - \infty\), and \(1^\infty\), special methods like l'H么pital's Rule are needed because substitution fails to provide valid results. Substitution works well for simple limits; however, it struggles to handle the undefined or ambiguous nature of indeterminate forms. L'H么pital's Rule effectively calculates these limits by finding the limit of the ratio of their derivatives, resulting in a correct and well-defined evaluation. An example of this is \(\lim_{x \rightarrow 0} \frac{\sin(x)}{x}\), where substitution results in \(\frac{0}{0}\), while l'H么pital's Rule successfully finds the limit to be \(1\).

Step by step solution

01

Define indeterminate forms and substitution

Indeterminate forms are expressions that have an undefined or ambiguous value. Some common indeterminate forms are \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0^\infty\), \(\infty - \infty\), and \(1^\infty\). Substitution is a method used to evaluate limits by replacing variables with specific values, usually as the variable approaches a certain number.
02

Explain the limitations of substitution

Substitution works well when evaluating simple limits, but it may fail when dealing with indeterminate forms. The reason is that indeterminate forms do not have a well-defined value鈥攕imply replacing variables with the values they approach may lead to incorrect or undefined results. Therefore, special methods like l'H么pital's Rule are sometimes needed to correctly evaluate these types of limits.
03

Introduce l'H么pital's Rule

L'H么pital's Rule is a method used to evaluate the limits of indeterminate forms \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\). It states that if the limit of a function with the form \(\frac{f(x)}{g(x)}\) is indeterminate, then the limit is equal to the limit of the ratio of their derivatives: $$\lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \lim_{x \rightarrow c} \frac{f'(x)}{g'(x)}$$ provided that the limit on the right-hand side exists or is \(\infty\) or \(-\infty\).
04

Provide an example of when l'H么pital's Rule is necessary

Let's look at an example to better understand why l'H么pital's Rule is needed for evaluating indeterminate forms: Evaluate the limit of \(\lim_{x \rightarrow 0} \frac{\sin(x)}{x}\).
05

Demonstrate the failure of substitution and the success of l'H么pital's Rule

Using substitution, we would directly replace \(x\) with \(0\): $$\lim_{x \rightarrow 0} \frac{\sin(x)}{x} = \frac{\sin(0)}{0} = \frac{0}{0}$$ This is an indeterminate form, and substitution fails to provide us with a valid answer. Now, let's try l'H么pital's Rule: The derivatives of the numerator and denominator are: $$f'(x) = \cos(x) \quad \text{and} \quad g'(x) = 1$$ So, applying l'H么pital's Rule: $$\lim_{x \rightarrow 0} \frac{\sin(x)}{x} = \lim_{x \rightarrow 0} \frac{\cos(x)}{1} = \frac{\cos(0)}{1} = 1$$ Hence, the limit is equal to \(1\), which is the correct answer. In this case, l'H么pital's Rule provided the necessary tool for evaluating the limit, when substitution could not.

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