Problem 9 - 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. $$\lim _{x \rightarrow 0^{+}} \frac{\ln x}{x}$$

Step-by-Step Solution

Answer

The limit is \(-\infty\).

Step 1: Determine the Form of the Limit

Evaluate the expression \( \frac{\ln x}{x} \) as \( x \rightarrow 0^+ \). Notice that as \( x \rightarrow 0^+ \), \( \ln x \rightarrow -\infty \) and \( x \rightarrow 0^+ \). Therefore, the limit takes the form \( \frac{-\infty}{0^+} \), which directly evaluates to \(-\infty\).

Step 2: Assess Appropriateness for l'Hospital's Rule

L'Hospital's Rule applies to limits of the form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Here, the form is \( \frac{-\infty}{0^+} \), which does not meet the criteria. Therefore, l'Hospital’s Rule is not applicable.

Key Concepts

l'Hospital's Rule

L'Hospital's Rule is a useful mathematical tool when evaluating the limits of indeterminate forms. It is specifically helpful when a limit results in forms like

• \( \frac{0}{0} \)
• \( \frac{\infty}{\infty} \)

When these forms appear, l'Hospital's Rule allows the substitution of the original function's derivatives. Instead of evaluating the original expression, you differentiate the numerator and the denominator separately. Find the new limit of these derivatives and continue with this until you reach a determinate form or exhaust the possibilities.

It is essential to evaluate the form of the limit before applying the rule. If your expression doesn't meet the criteria (like in the exercise where the form was \( \frac{-\infty}{0^+} \)), l'Hospital's Rule cannot be used. Instead, use other limit-solving techniques. Understanding when and how to apply l'Hospital's Rule will help simplify many complex limit problems.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the logarithm to the base \(e\). The number \(e\) is an irrational constant approximately equal to 2.71828. This function is crucial for analyzing how quantities grow exponentially. The natural logarithm's characteristic of converting multiplication into addition is handy for solving complex equations.

When assessing limits involving \( \ln(x) \), remember these key points:

• As \( x \rightarrow 0^+ \), \( \ln(x) \rightarrow -\infty \)
• As \( x \rightarrow \infty \), \( \ln(x) \rightarrow \infty \)
• \( \ln(1) = 0 \)

Having this knowledge helps evaluate expressions like \( \frac{\ln(x)}{x} \), especially for limits where \(x\) approaches boundaries like zero or infinity. Natural logarithms often appear in calculus problems, so understanding them can enhance problem-solving skills significantly.

Asymptotic Behavior

Asymptotic behavior describes how functions behave as their input approaches certain points (often infinity or zero). In the context of the given exercise, it's crucial to understand how expressions like \( \frac{\ln(x)}{x} \) behave as \( x \rightarrow 0^+ \). This understanding helps predict the function's limits and trends without complicated calculations.

In asymptotic analysis, terms like "tends toward" are common, reflecting the direction of behavior:

• An expression "approaches infinity" when values increase without bound.
• An expression "approaches zero" when values get infinitesimally small.

Considerations of asymptotic behavior are essential when evaluating limits and determining the action needed. For example, recognizing that \( \ln(x) \rightarrow -\infty \) when \( x \rightarrow 0^+ \) directly influenced the determination that the limit \( \frac{\ln(x)}{x} \) is \( -\infty \). Understanding these behaviors ensures accurate limit calculations and deeper insights into functional trends.