Problem 27 - Essential Calculus Early Transcendentals

Question

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 1^{+}} \ln x \tan (\pi x / 2)$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Analyze the expression

We are given the limit \( \lim _{x \rightarrow 1^{+}} \ln x \tan (\pi x / 2) \). We need to determine if it results in an indeterminate form, which is necessary for applying L'Hospital's Rule.

Step 2: Evaluate the behavior of each function component

As \( x \to 1^+ \), \( \ln x \to \ln 1 = 0 \) and \( \tan \left( \frac{\pi x}{2} \right) \to \tan \left( \frac{\pi}{2} \right) = +\infty \). Therefore, the expression has the form \( 0 \cdot \infty \). This is not a standard indeterminate form for L'Hospital's Rule.

Step 3: Rewrite the expression to use L'Hospital's Rule

Rewrite the expression \( \ln x \tan(\pi x / 2) \) as a quotient: \( \frac{\ln x}{\cot(\pi x / 2)} \). Now, both the numerator \( \ln x \to 0 \) and the denominator \( \cot(\pi x / 2) \to 0 \) as \( x \to 1^+ \), creating a \( \frac{0}{0} \) indeterminate form.

Step 4: Apply L'Hospital's Rule

Differentiate the numerator and denominator separately. The derivative of \( \ln x \) is \( \frac{1}{x} \), and the derivative of \( \cot(\pi x / 2) \) is \( -\frac{\pi}{2} \csc^2(\pi x / 2) \). Apply L'Hospital's Rule: \[\lim_{x \rightarrow 1^+} \frac{\ln x}{\cot(\pi x / 2)} = \lim_{x \rightarrow 1^+} \frac{\frac{1}{x}}{-\frac{\pi}{2} \csc^2(\pi x / 2)}\]

Step 5: Simplify the limit expression

Simplify the expression: \[\lim_{x \rightarrow 1^+} \frac{\frac{1}{x}}{-\frac{\pi}{2} \csc^2(\pi x / 2)} = \lim_{x \rightarrow 1^+} \frac{2}{-\pi x \csc^2(\pi x / 2)}\] As \( x \to 1^+ \), the limit evaluates to \( \frac{2}{-\pi} \cdot 0 = 0 \).

Step 6: Finalize the solution

Since the simplified limit evaluates to \( 0 \), the final answer is that the original limit \( \lim _{x \rightarrow 1^{+}} \ln x \tan (\pi x / 2) \) is 0.

Key Concepts

Indeterminate Forms

Indeterminate forms arise in calculus when the limit of a function evaluates to an uncertain or undefined expression. Common forms include \(\frac{0}{0}\), \(\infty\cdot0\), \(\frac{\infty}{\infty}\), \(\infty^0\), and \(1^\infty\).
These forms occur frequently when dealing with limits and require special techniques to resolve, such as L'Hospital's Rule.

• Example in this Exercise: Initially, the expression \(\ln x \tan(\pi x / 2)\) results in the form \(0 \cdot \infty\) as \(x\) approaches \(1^+\). However, \(0 \cdot \infty\) is not a standard indeterminate form that L'Hospital's Rule can directly handle.
• To manage this, we transformed it into a form suitable for L'Hospital's Rule by rewriting it as \(\frac{\ln x}{\cot(\pi x / 2)}\), leading to the \(\frac{0}{0}\) indeterminate form.

Limits in Calculus

Limits are a fundamental concept in calculus, describing the behavior of functions as they approach a specific point or value.
They help in understanding the tendency or potential result of functions in extreme scenarios, where direct evaluation isn't possible.

• The notion is vital because many functions may not have simple or obvious values at certain points but trend towards a distinct result.
• In the given exercise: We explore the limit as \(x\) approaches \(1^+\) for the function \(\ln x \tan(\pi x / 2)\).

We first assess each function component separately:

• \(\ln x\) approaches 0 as \(x\) nears 1.
• \(\tan(\pi x / 2)\) approaches \(\infty\) as \(x\) nears 1.

Combining these insights showed an apparent indeterminate form, leading us to further manipulation using L'Hospital's Rule.

Differentiation Techniques

Differentiation techniques are essential for simplifying indeterminate forms using L'Hospital's Rule.
This rule allows us to differentiate the numerator and denominator of a quotient separately to resolve the limit of indeterminate forms like \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\).
In this exercise, the differentiation steps were as follows:

• The function \(\ln x\) was differentiated to \(\frac{1}{x}\).
• The function \(\cot(\pi x / 2)\) required a bit more calculus knowledge. It was differentiated as \(-\frac{\pi}{2} \csc^2(\pi x / 2)\).

Applying L'Hospital's Rule involves substituting these derivatives back into the limit:

• We calculated the new limit: \(\lim_{x \rightarrow 1^+} \frac{\frac{1}{x}}{-\frac{\pi}{2} \csc^2(\pi x / 2)}\).

Finally, simplifying the resultant expression provides a definitive answer to the limit query, leading us to conclude that the limit is 0.