Problem 21 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(f(x)=\frac{\sin x}{x}\). This function will be quite important when we are interested in the derivative of sine and cosine. (a) What is the domain of \(f(x)\) ? (b) Use a graphing calculator or computer to help you sketch the graph of \(f(x)\). (c) Although \(f(x)\) is undefined at \(x=0, \lim _{x \rightarrow 0} f(x)\) exists. What do you think this limit might be? Check out your conjecture numerically. Observe that if \(\lim _{x \rightarrow 0} \frac{\sin x}{x}=L\), then for \(x\) very close to zero, \(\frac{\sin x}{x} \approx L\), or, equivalently, \(\sin x \approx L x\) for \(x\) close to zero.

Step-by-Step Solution

Answer

The domain of the function is all real numbers except 0. The function is not defined at \(x = 0\) but its limit as x approaches 0 does exist and is equal to 1. The function can be approximated by \(\sin x \approx x\) for \(x\) very close to zero.

Step 1: Determine the Domain of the Function

The domain of the function \(f(x) = \frac{\sin x}{x}\) is all real numbers except zero. This is because we can't divide by zero. So, \(x\) anything other than zero is in the domain.

Step 2: Graph the Function

Graph of the function \(f(x) = \frac{\sin x}{x}\) is required to visualize its behavior. The function oscillates between positive and negative values. It tends to zero as \(x\) goes to positive or negative infinity. The amplitude of the oscillations becomes smaller when \(x\) moves further from zero. Although, at \(x = 0\) the function is not defined, we could define \(f(0)\) to be the limit of \(f(x)\) as \(x\) approaches 0 to make the function continuous.

Step 3: Compute the Limit as x approaches 0

The limit of the function \(f(x) = \frac{\sin x}{x}\) as \(x\) approaches 0 is evaluated. It might seem that the limit is undefined because we have 0 in the denominator. However, L'Hopital's rule states that in such cases the limit can still be defined and is equal to the limit of the ratio of the derivatives of numerator and denominator. The derivative of \(\sin x\) with respect to \(x\) is \(\cos x\) and the derivative of \(x\) with respect to \(x\) is 1. Therefore, \(\lim _{x \rightarrow 0} \frac{\sin x}{x}\) is \(\lim _{x \rightarrow 0}\cos x =\cos 0 =1\). For \(x\) very close to zero, we can write \(\sin x \approx x\).

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial in calculus. The domain tells us all possible values of the independent variable, usually denoted by \(x\), that a function can accept. For the function given by \(f(x) = \frac{\sin x}{x}\), identifying its domain requires examining where the expression is defined. Since division by zero is undefined in mathematics, the point \(x = 0\) must be excluded from the domain of \(f(x)\).

Thus, the domain consists of all real numbers except zero, meaning \(x eq 0\).

• The function works fine for every other real number, as \(\sin x\) is defined everywhere.
• When assessing domains, be cautious of factors like square roots or denominators that could create undefined conditions.

By acknowledging this exclusion, we can better anticipate how the function behaves across its valid range.

Graphing Trigonometric Functions

Graphing trigonometric functions like \(\frac{\sin x}{x}\) helps us visualize their properties and behaviors. To sketch its graph, consider using graphing calculators or computer software to enhance accuracy. This offers a clear view of critical features such as oscillations and asymptotic behavior.

The function \(\frac{\sin x}{x}\) exhibits dampened oscillations because while \(\sin x\) oscillates between -1 and 1, dividing by an increasing \(x\) reduces the amplitude of the oscillations as \(x\) moves away from zero.

• The function approaches zero as \(x\) moves towards positive or negative infinity.
• At \(x = 0\), the function is undefined, but observing the graph shows it can approach a single point here through its limit.

By plotting these functions, we embrace both visual and numerical insights into their dynamic nature.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus used to find limits of indeterminate forms such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). This rule is particularly useful in evaluating the limit of \(\frac{\sin x}{x}\) as \(x\) approaches zero.

To apply L'Hôpital's Rule, take the derivatives of the numerator and denominator separately:

• The derivative of \(\sin x\) is \(\cos x\).
• The derivative of \(x\) is 1.

Thus, we compute the limit as \(x \rightarrow 0\) of \(\frac{\sin x}{x}\) by evaluating the limit of \(\frac{\cos x}{1}\), which simplifies to \(\cos 0 = 1\).
This result confirms that \(\lim_{x \to 0} \frac{\sin x}{x} = 1\), solidifying its importance in trigonometry and calculus. Always ensure functions meet conditions for applying this rule: both original limit forms must be indeterminate, and derivatives must exist.