Problem 87 - Essential Calculus

Question

Consider the function \(h(x)=\frac{x+\sin x}{x}\). (a) Use a graphing utility to graph the function. Then use the zoom and trace features to investigate \(\lim _{x \rightarrow \infty} h(x)\). (b) Find \(\lim _{x \rightarrow \infty} h(x)\) analytically by writing \(h(x)=\frac{x}{x}+\frac{\sin x}{x}\) (c) Can you use L'Hôpital's Rule to find \(\lim _{x \rightarrow \infty} h(x) ?\) Explain your reasoning.

Step-by-Step Solution

Answer

The limit of the function \(h(x)=\frac{x+\sin x}{x}\) as \(x\) approaches infinity is 1, determined both graphically and analytically. L'Hôpital's Rule cannot be used in this case to directly find the limit, as neither the numerator nor the denominator of the function approach zero or infinity as \(x\) approaches infinity.

Step 1: Graphical Analysis

On a graphing utility, plot the function \(h(x)=\frac{x+\sin x}{x}\). Use the zoom and trace features to closely observe the behavior of the function as \(x\) approaches infinity. The function should be seen to be converging towards a certain value. This gives a numerical estimate of the limit.

Step 2: Analytical Calculation

Express \(h(x)\) as the sum of two fractions: \(h(x)=\frac{x}{x} + \frac{\sin x}{x}\). The limit of the function as \(x\) approaches infinity is the sum of the limits of these two fractions. As \(x\) approaches infinity, \(\frac{x}{x}\) approaches 1, and \(\frac{\sin x}{x}\) approaches 0, since the sine function oscillates between \(-1\) and \(1\). So, \(\lim_{x \rightarrow \infty} h(x) = 1 + 0 = 1\).

Step 3: L'Hôpital's Rule Applicability

L'Hôpital's Rule can be used to find a limit if the function can be expressed in the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), and the derivative of the numerator and the denominator can be calculated. In this case, the function \(h(x)\) does not satisfy either of these conditions, as neither the numerator \(x + \sin(x)\) nor the denominator \(x\) approach zero or infinity as \(x\) approaches infinity. Consequently, L'Hôpital's Rule cannot be applied to directly find the limit of this function as \(x\) approaches infinity.

Key Concepts

Using Graphing Utilities to Visualize Limits

When working with calculus and investigating limits, graphing utilities can be invaluable tools. These are software or calculator functions that allow you to plot graphs of mathematical functions and analyze their behavior. By using a graphing utility to plot the function \( h(x)=\frac{x+\sin x}{x} \), students can visually observe the trend of the function as \( x \) approaches infinity.

• Graphing utilities often have zoom and trace features. These allow you to look closely at portions of the graph and follow the "trace" of the function, making it easier to predict where the function is heading as \( x \) increases.
• For this function, a graphing utility can show how the function seems to approach a horizontal line, suggesting the limit value is stable as \( x \) increases.

Graphical analysis provides a numerical estimate of the limit, offering a visual verification before proceeding to analytical methods.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus for finding limits, particularly when dealing with indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
This rule states that if the limit of a ratio of two functions results in these forms, the limit of the ratio is equal to the limit of the ratio of their derivatives, provided the derivatives themselves have limits.However, in the case of the function \( h(x) = \frac{x + \sin x}{x} \), we cannot use L'Hôpital's Rule.

• In order to apply L'Hôpital's Rule, the expression needs to be either in \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) form. For \( h(x) \), neither condition is met as \( x \) tends towards infinity.
• The numerator \( x + \sin(x) \) does not become indeterminate relative to the denominator \( x \); instead, it's dominated by \( x \), simplifying the limit more directly through its components.

This directs us instead to employ alternative methods, like simplifying the function directly, to find the limit at infinity.

Understanding Asymptotic Behavior

Asymptotic behavior describes how a function behaves as it approaches a particular point or infinity. In the context of this problem, we're interested in the asymptotic behavior of the function \( h(x) = \frac{x + \sin x}{x} \) as \( x \to \infty \).

• An asymptote is a line that a graph approaches but never touches. As you look at the graph of \( h(x) \), you'll notice it seems to converge to a straight line as \( x \) increases. This horizontal line is the horizontal asymptote.
• The function can be rewritten as \( h(x) = 1 + \frac{\sin x}{x} \). Here, \( \frac{\sin x}{x} \) becomes negligible as \( x \) goes to infinity because \( \sin x \) is bounded between -1 and 1.
• Thus, \( h(x) \) approaches 1, indicating its asymptotic behavior is horizontal along the line \( y=1 \).

Recognizing this pattern is essential for understanding the limit behavior of functions as they extend towards infinity.