Problem 60 - Calculus - AP Edition

Question

Verify that L'Hôpital's rule is of no help in finding the limit; then find the limit, if it exists, by some other method. $$\lim _{x \rightarrow+\infty} \frac{2 x-\sin x}{3 x+\sin x}$$

Step-by-Step Solution

Answer

The limit is \(\frac{2}{3}\).

Step 1: Check the form of the limit

First, check if the limit is of an indeterminate form. Substitute \[ \lim _{x \rightarrow+\infty} \frac{2x - \sin x}{3x + \sin x} \]Both the numerator, \(2x - \sin x\), and the denominator, \(3x + \sin x\), approach infinity as \(x\) approaches infinity. Thus, the limit is of the form \(\frac{\infty}{\infty}\), which suggests that L'Hôpital's rule could potentially be used. However, we'll try another method as per the exercise's instructions.

Step 2: Simplify the expression by dividing by the highest term

Divide both the numerator and the denominator by \(x\), the highest power of \(x\) present in the original expression:\[\frac{\frac{2x}{x} - \frac{\sin x}{x}}{\frac{3x}{x} + \frac{\sin x}{x}}\]This simplifies to:\[\frac{2 - \frac{\sin x}{x}}{3 + \frac{\sin x}{x}}\]

Step 3: Analyze the behavior of \(\frac{\sin x}{x}\) as \(x\) approaches infinity

As \(x\) approaches infinity, the term \(\frac{\sin x}{x}\) tends to 0 because \(\sin x\) oscillates between -1 and 1, while \(x\) becomes infinitely large. The term \(\frac{\sin x}{x}\) becomes negligible.

Step 4: Evaluate the limit

Substitute the results from Step 3 into the simplified expression:\[\lim _{x \rightarrow+\infty} \frac{2 - \frac{\sin x}{x}}{3 + \frac{\sin x}{x}} = \frac{2 - 0}{3 + 0} = \frac{2}{3}\] Therefore, the limit exists and is \(\frac{2}{3}\).

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a powerful mathematical tool that helps us evaluate limits involving indeterminate forms such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). The rule states that if a limit initially results in one of these indeterminate forms, you can differentiate the numerator and the denominator separately, and then take the limit again. In simpler terms, it's a way to find a limit when direct substitution leads us nowhere because both the top and bottom parts of a fraction go to zero or infinity. L'Hôpital's Rule is like a shortcut. Instead of dealing with complex infinite or zero values, you take the derivative, which might simplify the process. However, sometimes L'Hôpital's Rule may not be beneficial. As in the given problem, even though the limit \(\lim _{x \rightarrow+\infty} \frac{2x - \sin x}{3x + \sin x}\) initially appears to be an \(\frac{\infty}{\infty}\) form, using the rule is not always the best route. The problem specifically instructs us to use an alternative method, showing that while useful, L'Hôpital's Rule is not always the necessary solution.

Indeterminate Forms

Indeterminate forms pop up when evaluating limits, suggesting a lack of clear, immediate value. These forms include \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(\infty - \infty\), \(0 \cdot \infty\), \(1^\infty\), \(0^0\), and \(\infty^0\). Each represents situations where substitution in the function does not directly yield a result, creating a kind of mathematical ambiguity. When you find yourself facing an indeterminate form, it often signals the need for further manipulation or analysis to properly evaluate the limit. For instance, recognizing the original problem as an \(\frac{\infty}{\infty}\) indeterminate form suggests it's a candidate for techniques like L'Hôpital's Rule or algebraic manipulations, like dividing components by the highest degree of \(x\) to simplify the expression. In the problem, recognizing that \(\lim _{x \rightarrow+\infty} \frac{2x - \sin x}{3x + \sin x}\) was \(\frac{\infty}{\infty}\) led us to simplify the expression rather than apply L'Hôpital's Rule. This step was crucial to evaluating the limit without being misled by the initial indeterminate form.

Asymptotic Behavior

Asymptotic behavior describes how functions behave as they move towards infinity or a particular point, but never exactly get there. Think of it like trailing but never actually reaching. This behavior is critical when evaluating limits, as it helps in understanding what the function approaches in the extreme ends. In analyzing the asymptotic behavior of \(\frac{2x - \sin x}{3x + \sin x}\), we divided every term by \(x\), which simplified the expression. By observing how terms like \(\frac{\sin x}{x}\) behave as \(x\) approaches infinity, where \(\sin x\) remains bounded between -1 and 1, the asymptotic nature of \(\frac{\sin x}{x}\) shows it heads towards zero. Thus, the impactful observation here is understanding the larger components \(2x\) and \(3x\) dictate the asymptotic behavior. The strategy simplifies complex terms, thus pushing non-influential terms towards zero as \(x\) grows. This asymptotic insight led us to conclude that the limit evaluates to \(\frac{2}{3}\), showcasing how powerful understanding asymptotic behavior can truly be.