Problem 38 - Calculus for Biology and Medicine

Question

Let $$ f(x)=\frac{\sin x}{x}, \quad x \neq 0 $$ (a) Show that \(y=0\) is a horizontal asymptote. (b) Since \(f(x)\) is not defined at \(x=0\), does this mean that \(f(x)\) has a vertical asymptote at \(x=0\) ? Find \(\lim _{x \rightarrow 0^{+}} f(x)\) and \(\lim _{x \rightarrow 0^{-}} f(x) .\) (c) Use a graphing calculator to sketch the graph of \(f(x)\).

Step-by-Step Solution

Answer

(a) Yes, \( y=0 \) is a horizontal asymptote. (b) No vertical asymptote; \( \lim_{x \to 0^{+}} f(x) = \lim_{x \to 0^{-}} f(x) = 1.\) (c) Graph confirms \( y=0 \) as a horizontal asymptote, with no vertical asymptote.

Step 1: Identify the horizontal asymptote

For a function \( f(x) \) to have a horizontal asymptote at \( y = L \), the limit \( \lim_{x \to \infty} f(x) = L \) must hold. We need to compute the limit \( \lim_{x \to \infty} \frac{\sin x}{x} \). As \( x \to \infty \), \( \sin x \) oscillates between -1 and 1, hence \( \frac{-1}{x} \leq \frac{\sin x}{x} \leq \frac{1}{x} \). The limits on the right and left both approach 0. Thus, by the Squeeze Theorem, \[\lim_{x \to \infty} \frac{\sin x}{x} = 0\.\] Therefore, \( y = 0 \) is a horizontal asymptote.

Step 2: Evaluate potential vertical asymptote

To check for a vertical asymptote at \( x = 0 \), the limits \( \lim _{x \rightarrow 0^{+}} f(x) \) and \( \lim _{x \rightarrow 0^{-}} f(x) \) must approach \( \pm \infty \). However, \( f(x) = \frac{\sin x}{x} \) can be evaluated using the limit \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \). Therefore, \[\lim_{x \to 0^{+}} \frac{\sin x}{x} = 1\] and \[\lim_{x \to 0^{-}} \frac{\sin x}{x} = 1\.\] Since both limits yield the finite number 1, \( f(x) \) does not have a vertical asymptote at \( x = 0 \).

Step 3: Graphing the function

Using a graphing calculator, plot \( f(x) = \frac{\sin x}{x} \). Observe that the graph approaches the x-axis (\( y = 0 \)) as \( x \to \infty \) and \( x \to -\infty \), confirming the horizontal asymptote. Near \( x = 0 \), the graph smoothly passes through \( (0,1) \) due to the limit calculated earlier, and there is no vertical asymptote.

Key Concepts

Horizontal Asymptote

Horizontal asymptotes help us understand the behavior of a function as the input, or \( x \), becomes very large or very small. They represent a line that the graph of the function approaches but never quite touches as \( x \) goes to infinity or negative infinity.
In our specific example, we have a function \( f(x) = \frac{\sin x}{x} \). To find its horizontal asymptote, we analyze what happens as \( x \to \infty \).
Since \( \sin x \) oscillates between -1 and 1, the ratio \( \frac{\sin x}{x} \) results in smaller and smaller fractions, effectively approaching zero.
This outcome is confirmed using the Squeeze Theorem, and hence, the horizontal asymptote is \( y = 0 \).

• As \( x \to \infty \), \( \lim_{x \to \infty} \frac{\sin x}{x} = 0 \)
• The graph gets closer to but does not touch the line \( y = 0 \).

Vertical Asymptote

Vertical asymptotes differ from horizontal asymptotes as they indicate the \( x \)-value(s) where a function becomes infinite. Mathematically, this happens when the limit of \( f(x) \) approaches \( \pm \infty \) as \( x \) approaches a particular value.
In this example, we tested for a vertical asymptote at \( x = 0 \) for the function \( f(x) = \frac{\sin x}{x} \).
By evaluating the limits from the right and left at \( x = 0 \), where \( \lim_{x \to 0^{+}}\) and \( \lim_{x \to 0^{-}} \), we find that both limits equal 1.
This finite value indicates there is no vertical asymptote at \( x = 0 \), and the function smoothly crosses the \( y \)-axis at this point.

• A function with a finite limit at \( x \to 0 \) means no vertical asymptote exists.
• \( \lim_{x \to 0^{+}} \frac{\sin x}{x} = 1 \) and \( \lim_{x \to 0^{-}} \frac{\sin x}{x} = 1 \)

Squeeze Theorem

The Squeeze Theorem is a method in calculus to find the limit of a function. It's especially useful when direct evaluation is challenging or when the function is oscillating.
If you have three functions, \( g(x) \), \( f(x) \), and \( h(x) \), and \( g(x) \leq f(x) \leq h(x) \), then if \( \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L \), it follows that \( \lim_{x \to a} f(x) = L \) as well.
In our example, the Squeeze Theorem is applied to \( \frac{\sin x}{x} \). Since \( -1 \leq \sin x \leq 1 \), we have:

• \( \frac{-1}{x} \leq \frac{\sin x}{x} \leq \frac{1}{x} \)

These limits can be squared down to find \( \lim_{x \to \infty} \frac{\sin x}{x} = 0 \).
So, the Squeeze Theorem elegantly shows us that \( f(x) \) approaches 0 at infinity.

Limits

Limits are a fundamental concept in calculus that signify a value that a function approaches as the variable gets close to some point.
In this exercise, limits help us determine the behavior of the function \( f(x) = \frac{\sin x}{x} \) as \( x \) approaches infinity and zero.
By evaluating \( \lim_{x \to 0} \), we find that the important limit \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \) demonstrates that the function does not have a vertical asymptote at zero, but passes through this point with continuity.

• Limits can verify whether a function approaches a specific value or infinity.
• They're crucial in analyzing differences, derivatives, and for identifying asymptotic behaviors.

Graphing Functions

Graphing functions visually represents the behavior and essential characteristics of mathematical equations. It is a powerful tool to understand the asymptotic behavior by seeing how a function behaves as \( x \) approaches different values.
In our example of \( f(x) = \frac{\sin x}{x} \), a graphing calculator shows us the curve's essential features such as:

• How the function approaches the horizontal line \( y = 0 \) as \( x \to \infty \) and \( x \to -\infty \).
• Despite not being defined at \( x = 0 \), the graph's continuity at that point with the calculated limit ensuring a smooth transition.

Graphing is an effective means to verify theoretical insights such as limits and asymptotes.