Question

Graphical, Numerical, and Analytic Analysis In Exercises \(75-82\) , use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{x \rightarrow 0} \frac{\sin x}{\sqrt[3]{x}} $$

Short answer

The graphical and numerical investigation suggests a certain limit. The exact limit will be calculated analytically by applying L'Hopital's rule and replacing the x-values in the derivative equation.

Step-by-step solution

Graphical Analysis

To use a graphing utility (like GeoGebra or Desmos), plot the function \( y = \frac{\sin x}{\sqrt[3]{x}} \). You are likely to see the graph is approaching a certain value as x gets closer to 0. This is your initial estimate of the limit, which will help you to cross-check the final analytical result.

Numerical Analysis

A table of function values could be prepared for values very close to 0, both from the right and left side. For example, evaluate the function \( y = \frac{\sin x}{\sqrt[3]{x}} \) at x = -0.1, -0.01, 0.01, 0.1. These function values should seem to approach a certain number as x gets closer to 0. This estimated limit should match the graphical estimate.

Analytical Analysis

For analytical solution, L'Hopital's rule, which says that the limit of a quotient of two functions at a point where both approach zero is equal to the limit of the quotients of their derivatives, can be applied. For \( f(x) = \sin x \) and \( g(x) = \sqrt[3]{x} \), find derivatives \( f'(x) = \cos x \) and \( g'(x) = \frac{1}{3\sqrt[3]{x^2}} \). The limit then becomes \( \lim _{x \rightarrow 0} \frac{f '(x)}{g'(x)} =\lim _{x \rightarrow 0} \frac{\cos x}{\frac{1}{3\sqrt[3]{x^2}}}\). Taking this limit as x approaches 0 gives the exact limit.

Compare Results and Validate Limit

Compare the limit obtained analytically with the results obtained from the graph and numerical table. If all three are consistent, you can conclude that the limit calculated is correct.

Key concepts

Graphical Analysis

Graphical analysis helps visually estimate the behavior of functions as they approach a certain value. For the limit \( \lim _{x \rightarrow 0} \frac{\sin x}{\sqrt[3]{x}} \), a graphing tool can provide a clear picture of how this function behaves near \(x = 0\). By plotting the function with tools like GeoGebra or Desmos:

• You will see the curve approaching a specific value as \(x\) gets closer to zero.
• This visual indication is an initial estimate of the limit.

Using this step, you ensure your analytical result is cross-verified by what you see graphically. Observing the plot can reveal any sudden changes or unusual behavior that might be missed otherwise.

Numerical Analysis

Numerical analysis provides a practical way of backing up graphical estimations. By calculating the function values at points near zero, you enhance your understanding of the function's limit.

• Create a table with values of \(x\) such as \(-0.1, -0.01, 0.01, 0.1\).
• Calculate the function \( y = \frac{\sin x}{\sqrt[3]{x}} \) at these points.
• Notice how the output values tend toward a certain number as \(x\) approaches zero.

This step strengthens the estimate obtained from the graph, confirming consistency. When both graphical and numerical results suggest the same limit, confidence in the estimate increases.

L'Hopital's Rule

L'Hopital's Rule provides an analytical method for finding limits, particularly useful when dealing with indeterminate forms like \( \frac{0}{0} \). In our function, apply L'Hopital's Rule on

• \( f(x) = \sin x \), with the derivative \( f'(x) = \cos x \).
• \( g(x) = \sqrt[3]{x} \), with the derivative \( g'(x) = \frac{1}{3\sqrt[3]{x^2}} \).

Recalculate the limit as:\[\lim _{x \rightarrow 0} \frac{\cos x}{\frac{1}{3\sqrt[3]{x^2}}}\]Taking the new limit as \(x\) approaches 0 with derivatives allows us to bypass initial indeterminacy, leading to a clear numerical result. This powerful rule often simplifies complex limits into manageable calculations.

Analytical Methods

Analytical methods involve detailed mathematical procedures to precisely determine limits. Examining the limit \( \lim _{x \rightarrow 0} \frac{\sin x}{\sqrt[3]{x}} \) through these methods:

• Requires evaluating the fundamental behavior of the function components as \(x\) approaches zero.
• Often involves calculus concepts like derivatives, especially in cases of indeterminate forms such as \( \frac{0}{0} \).
• Validating results from graphical and numerical analyses enriches understanding and confirms correctness.

When the analytical, numerical, and graphical perspectives align, the confidence in your solution is maximized. This multifaceted combination ensures a comprehensive grasp of the limit and its calculation.