Question

Use a calculator to graph the function and estimate the value of the limit, then use L'Hôpital's rule to find the limit directly. $$ \lim _{x \rightarrow 0^{+}} \frac{\ln x}{\sin x} $$

Short answer

The limit is \(-\infty\).

Step-by-step solution

Understand the Problem

We are given the limit expression \( \lim _{x \rightarrow 0^{+}} \frac{\ln x}{\sin x} \). Our task is to estimate the limit using a calculator graph and then confirm by applying L'Hôpital's rule.

Graph the Function

Use a graphing calculator to plot the function \( y = \frac{\ln x}{\sin x} \). Observe the behavior of the graph as \( x \) approaches \( 0^+ \). Note if the graph trends towards a specific value, which would be our estimated limit.

Estimate the Limit

From the graph, observe that as \( x \to 0^+ \), \( \ln x \to -\infty \) and \( \sin x \to x \), making \( \frac{\ln x}{\sin x} \to -\infty \). Thus, our estimate is that the limit approaches \(-\infty\).

Analyze using L'Hôpital's Rule

Recognize that the expression \( \frac{\ln x}{\sin x} \) is in an indeterminate form \( \frac{-\infty}{0} \) as \( x \rightarrow 0^{+} \). Thus, L'Hôpital's Rule, applicable in \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) forms, cannot directly apply. However, we observe \( \lim_{x \rightarrow 0^+} \ln x = -\infty \) and calculate derivatives for further insight if needed.

Confirm the Analysis

Despite the direct inapplicability of L'Hôpital's Rule here, consider the trends contractually: derivative of \( \ln x \) is \( \frac{1}{x} \) and that of \( \sin x \) is \( \cos x \). With this and the graphical insight, establish that as \( x \to 0^+ \), the original expression leads the function towards \( -\infty \).

Key concepts

Limits

Limits help us understand the behavior of functions as they approach a particular point. In the case of the function \( \lim_{x \rightarrow 0^{+}} \frac{\ln x}{\sin x} \), we aim to determine what happens to the expression as \( x \) gets arbitrarily close to zero from the positive side.
As \( x \) approaches 0 from the right, \( \ln x \) moves towards \(-\infty\), and \( \sin x \) approaches \( x \) itself.
Thus, the expression \( \frac{\ln x}{\sin x} \) comes to resemble \( \frac{-\infty}{0} \), which typically suggests the result tends towards \(-\infty\).

To solve limits involving complex expressions, sometimes intuition or numerical estimates from graphical outputs can be helpful. Observing the behavior of the function on a graph as \( x \) approaches zero can give an excellent visual cue to its behavior.

Graphing Functions

Graphing functions allows us to visualize how they behave, especially as they approach limits. For our function \( y = \frac{\ln x}{\sin x} \), graphing helps in estimating the limit by observing the trend as \( x \to 0^+ \).
Using a graphing calculator, you can directly see that \( \ln x \), which decreases quickly to \(-\infty\), overpowers \( \sin x \), which approaches zero. The graph would reveal a sharp downward trail, indicating that the function's values plunge steeply downward as \( x \) nears zero.

This visual analysis confirms the limit approaches \(-\infty\), which helps guide us even if the algebraic techniques alone seem tricky. Graphs essentially provide a clear confirmation and a practical approach to examining limits.

Indeterminate Forms

Indeterminate forms in calculus signal expressions where traditional algebra might fail to directly pinpoint the limit. In the expression \( \frac{\ln x}{\sin x} \) as \( x \to 0^+ \), the form \( \frac{-\infty}{0} \) arises.
Though it is technically not one of the classic indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), it poses a challenge for direct evaluation. Even though L’Hôpital’s Rule cannot directly apply here, understanding derivative behavior can be insightful.
Knowing that \( \ln x \) drastically falls towards \(-\infty\) and \( \sin x \approx x \), with its derivative \( \cos x \approx 1 \) at \( x=0 \), we infer the function's approach to \(-\infty\).

Addressing such indeterminate forms often involves recognizing the dominating function in the numerator or denominator, which in this scenario is \( \ln x \). The identification of these characteristics is crucial for evaluating challenging limits.