Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0}(\cos x)^{1 / x} $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify the Form

First, we need to determine the form of the given limit. Notice that as \(x\) approaches 0, \(\cos x\) approaches \(\cos(0) = 1\). Therefore, the expression \((\cos x)^{1/x}\) approaches \(1^\infty\), which is an indeterminate form.

Step 2: Transform the Expression

To apply l'Hôpital's Rule, we should transform the expression into a quotient. By taking the natural logarithm, we can rewrite the limit as \(\lim_{x \rightarrow 0} \frac{\ln((\cos x)^{1/x})}{1}\). Simplify to \(\lim_{x \rightarrow 0} \frac{1}{x} \ln(\cos x)\).

Step 3: Use L'Hôpital's Rule

The expression \(\ln(\cos x)\) approaches \(\ln(1) = 0\) as \(x\) approaches 0, and \(1/x\) approaches infinity, making \(\frac{\ln(\cos x)}{x}\) a \(\frac{0}{0}\) form, suitable for l'Hôpital's Rule. Differentiate the numerator and the denominator. The derivative of \(\ln(\cos x)\) is \(-\tan(x)\), and the derivative of \(x\) is 1.

Step 4: Evaluate the New Limit

Using l'Hôpital's Rule, we find \(\lim_{x \rightarrow 0} \frac{\ln(\cos x)}{x} = \lim_{x \rightarrow 0} \frac{-\tan x}{1}\). As \(x\) approaches 0, \(\tan x\) approaches 0, so \(\lim_{x \rightarrow 0} -\tan x = 0\).

Step 5: Apply the Exponential Function

Since we originally took the natural logarithm, we must exponentiate the result to find the original limit: \(\lim _{x \rightarrow 0} (\cos x)^{1/x} = e^0 = 1\).

Key Concepts

Indeterminate Forms

In calculus, certain limit expressions can lead to what are called indeterminate forms. These forms occur when substituting a value into a function does not yield a clear numerical result, indicating that the limit may need additional techniques for evaluation. The classic indeterminate forms involve operations such as division, multiplication, and exponentiation that yield results like \( \frac{0}{0} \), \( 0^0 \), or \( 1^\infty \). For example, in the expression \((\cos x)^{1/x}\) as \(x\) approaches 0, we encounter the indeterminate form \(1^\infty\). This means that directly substituting \(x = 0\) into the expression gives a form that is not straightforward to evaluate. To resolve such forms, calculus provides powerful tools like l'Hôpital's Rule, which aids in evaluating limits when these problematic expressions arise. Understanding indeterminate forms is crucial in the process of limit evaluation, as they often dictate the methods used to find solutions.

Limit Evaluation

Limit evaluation is a fundamental concept in calculus used to understand the behavior of functions as they approach certain values. When a function does not provide a clear result through direct substitution, we must employ different strategies to find its limit.In our example, the goal is to determine \( \lim_{x \rightarrow 0} (\cos x)^{1/x} \). Direct substitution yields \(1^\infty\), which is indeterminate. By recognizing this indeterminate form, we can proceed with methods like transformations or applying l'Hôpital's Rule to evaluate the limit properly.Transforming the expression is often a necessary step. In this case, taking the natural logarithm of the function helps us convert the limit into a more manageable form. Transformations allow us to rewrite expressions in a way that makes it easier to apply calculus techniques and find the correct limit.

Calculus Problem Solving

Solving calculus problems often involves determining limits, derivatives, or integrals of functions under various circumstances. Each problem presents unique challenges and requires a strategic approach to uncover solutions.For limits involving indeterminate forms such as \( (\cos x)^{1/x} \), problem solving involves several steps:

• Identify the form of the limit, ensuring it is indeterminate.
• Apply transformations like the natural logarithm to restructure the expression.
• Utilize l'Hôpital's Rule if the transformed expression presents a \( \frac{0}{0} \) form.
• Simplify the expression further if needed and take additional limits.
• Remember to reverse transformations, such as applying the exponential function, to return to the original limit.

By using these steps, calculus students can systematically approach and solve limit problems with clarity and precision.

Natural Logarithm Transformation

The natural logarithm transformation is a technique often used to make complex limit evaluations more manageable. When dealing with indeterminate forms, taking the natural logarithm can simplify the expression by converting products and powers into sums, which are typically easier to handle.In the problem \( \lim_{x \rightarrow 0} (\cos x)^{1/x} \), applying the natural logarithm leads to the equation \( \ln((\cos x)^{1/x}) = \frac{1}{x} \ln(\cos x) \). This transformation turns an exponential form into a quotient, allowing us to apply l'Hôpital's Rule confidently.The natural logarithm acts as a tool to break down the expression into a more straightforward component, so methods such as differentiation and limit laws can be applied effectively. This transformation is essential in calculus, especially when solving limits that initially appear complicated or undefined.