Problem 29 - Essential Calculus

Question

In Exercises \(27-38,\) (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L'Hôpital's Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow \infty}\left(x \sin \frac{1}{x}\right)\)

Step-by-Step Solution

Answer

The given limit \(\lim _{x \rightarrow \infty}\left(x \sin \frac{1}{x}\right)\) equals 1.

Step 1: Identify the Indeterminate Form

Start by substituting the limiting value into the function. Here, as \(x \rightarrow \infty\), \(\sin(1/x) \rightarrow \sin(0) = 0\) and \(x \rightarrow \infty\). So, the form of the limit is of type \(0 \times \infty\), an indeterminate form.

Step 2: Modify the Form to Apply L'Hôpital's Rule

You are required to modify this indeterminate form to either \(0/0\) or \(\infty /\infty\) to use L'Hôpital's Rule. Rewrite \(x \sin (1/x)\) as \(\frac{\sin (1/x)}{1/x}\) . Now, as \(x \rightarrow \infty\), both numerator and denominator tend to zero, hence it is of the form \(0/0\). It's now in the proper form for applying L'Hôpital's Rule.

Step 3: Apply L'Hôpital's Rule

L'Hôpital's Rule states that the limit of the ratio of two functions, where both go to zero or both go to infinity, is equal to the limit of the ratio of their derivatives. Proceed to apply this rule here. The derivative of the numerator \(\sin(1/x)\) is \(\cos(1/x)*(-1/x^2)\), and the derivative of the denominator \(1/x\) is \(-1/x^2\). Our limit now becomes \(\lim_{x\rightarrow\infty} \frac{\cos(1/x)*(-1/x^2)}{-1/x^2}\). Since \(-1/x^2\) is common in both numerator and denominator, it can be cancelled out.

Step 4: Evaluate the Final Limit

The limit now simplifies to \(\lim_{x\rightarrow\infty} \cos(1/x)\). Since as \(x \rightarrow \infty\), \(1/x \rightarrow 0\), this further simplifies to \(\cos(0)\), which equals 1.

Step 5: Graphing the Function

The graphical representation isn't retrievable in this format. To visualize, use a graphing utility and graph the function \(x \sin(1/x)\). As \(x \rightarrow \infty\), this function approaches 1, which confirms our calculated limit is correct.

Key Concepts

Indeterminate Forms

Indeterminate forms arise when evaluating limits where substitution results in undefined expressions. These forms often involve combinations such as \(0 \times \infty\), \( \frac{0}{0} \), or \( \frac{\infty}{\infty} \). Recognizing them is crucial because they indicate that further analysis is needed to find the limit. In our example, as \(x \rightarrow \infty\), the expression \(x \sin(1/x)\) directly leads to the indeterminate form \(0 \times \infty\). This is because \(\sin(1/x)\) approaches 0, and \(x\) heads towards infinity. Transforming this into a form suitable for L'Hôpital's Rule typically involves rewriting the expression to \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) to make the limit solvable.

Limits

Limits explore the behavior of functions as they approach specific points or infinity. In this context, limits help determine what the function value approaches as the input grows without bound. For expressions arriving at indeterminate forms, L'Hôpital's Rule provides a method to evaluate these limits. This rule applies when the limit results in \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), by differentiating the numerator and denominator separately and then taking the limit of their quotient. In our case, we rewrote \(x \sin(1/x)\) as \( \frac{\sin(1/x)}{1/x} \), transforming the limit to \( \frac{0}{0} \), which is appropriate for L'Hôpital's Rule. Consequently, we determined the limit as \( x \rightarrow \infty \) of \(x \sin(1/x)\) is 1.

Graphing Functions

Graphing functions visually verifies the behavior expected from algebraic analysis. It's particularly useful for complex functions or indeterminate forms, where the algebraic limit may not be intuitive. Using a graphing utility for the function \(x \sin(1/x)\), we observe how it behaves as \(x\) approaches infinity. The graph clearly shows the function trending towards a horizontal asymptote at 1, confirming our analytical result found using L'Hôpital's Rule. This visual reassurance affirms our understanding and provides a solid way to interpret the function's limit as \(x \rightarrow \infty\). Graphs also help highlight key characteristics of the function, such as oscillations or increasing behaviors, that analytically calculated limits might not easily reveal.