Question

Plot the numerator \(f(x)\) and the denominator \(g(x)\) in the same graph window for each of these domains: \(-1 \leq x \leq 1,-0.1 \leq x \leq 0.1\), and \(-0.01 \leq x \leq 0.01\). From the plot, estimate the values of \(f^{\prime}(x)\) and \(g^{\prime}(x)\) and use these to approximate the given limit. $ \lim _{x \rightarrow 0} \frac{3 x-\sin x}{x} $$

Short answer

The limit is 2.

Step-by-step solution

Define the Functions

First, identify the numerator and the denominator of the given expression. The numerator is \( f(x) = 3x - \sin x \), and the denominator is \( g(x) = x \).

Plot the Functions for Domain 1

For the domain \(-1 \leq x \leq 1\), plot the functions \( f(x) = 3x - \sin x \) and \( g(x) = x \) on the same graph. Observe the behavior of the functions as they approach zero.

Plot the Functions for Domain 2

For the domain \(-0.1 \leq x \leq 0.1\), plot the functions \( f(x) = 3x - \sin x \) and \( g(x) = x \) again. Zooming in further on the domain, observe the similarity in slopes around zero.

Plot the Functions for Domain 3

For the domain \(-0.01 \leq x \leq 0.01\), plot the functions \( f(x) = 3x - \sin x \) and \( g(x) = x \) again. This provides an even closer view near zero to better estimate the slope or derivative behavior of the functions.

Estimate Derivatives

Estimate the derivative \( f'(x) = 3 - \cos x \) at \( x=0 \), which is \( f'(0) = 3 - \cos(0) = 2 \). For \( g(x) = x \), the derivative \( g'(x) \) is 1 everywhere.

Apply L'Hôpital's Rule

Since both the numerator and denominator approach zero as \( x \to 0 \), apply L'Hôpital's Rule: \( \lim_{x \to 0} \frac{f(x)}{g(x)} = \lim_{x \to 0} \frac{f'(x)}{g'(x)} \). Substitute the estimated derivatives: \( \lim_{x \to 0} \frac{2}{1} = 2 \).

Key concepts

Limits in Calculus

Limits help us understand the behavior of functions as they get very close to a specific point. In this exercise, we are looking at the limit of the expression \( \lim _{x \rightarrow 0} \frac{3x-\sin x}{x} \). When approaching limits, we want to know what value the expression is getting close to at that particular point.
This usually involves finding out what happens when the input of the function approaches a certain value, such as zero in this case. Calculating limits is essential in calculus as it lays the groundwork for defining continuity and calculating derivatives. It allows us to precisely understand how a function behaves near a particular point without necessarily being defined at that point.

Derivatives

Derivatives are a central concept in calculus, used to describe how a function changes as its input changes. In simpler terms, a derivative represents the slope or steepness of a function's curve at a particular point.
For this exercise, the derivatives we calculate are \( f'(x) = 3 - \cos x \) and \( g'(x) = 1 \) at \( x = 0 \). The derivative of \( f(x) \) at this point gives us \( f'(0) = 2 \), indicating how rapidly the function \( f(x) \) is changing as \( x \) approaches zero. For the denominator function \( g(x) = x \), the derivative \( g'(x) = 1 \) remains constant, showing a steady rate of change.

Graphing Functions

Graphing allows us to visually comprehend and compare how functions behave across specific intervals or domains. When we graph the numerator \( f(x) = 3x - \sin x \) and denominator \( g(x) = x \), we use different zoom levels: from a broad view within \(-1 \leq x \leq 1 \), to a closer look \(-0.1 \leq x \leq 0.1 \), and finally an even tighter focus \(-0.01 \leq x \leq 0.01 \).
This step-by-step zooming on the graph helps us observe the similarity in the slopes and how these functions interact around the point of interest, which is zero. By visually estimating the rate of change or slope of these functions, it simplifies the process of calculating the derivatives and ultimately understanding their behavior.

Calculus Techniques

A common calculus technique used to solve indeterminate forms when finding limits is L'Hôpital's Rule. This rule comes in handy when both the numerator and denominator approach zero or infinity.
In the exercise, as \( x \rightarrow 0 \), both \( 3x - \sin x \) and \( x \) approach zero, which makes direct evaluation impractical. Instead, by applying L'Hôpital's Rule, we switch to using derivatives: \( \lim _{x \to 0} \frac{f'(x)}{g'(x)} \). This simplifies evaluating the original limit problem. Substituting the derivatives we calculated earlier, \( \lim _{x \to 0} \frac{2}{1} = 2 \), gives us a clear and manageable solution. L'Hôpital's Rule is an invaluable tool for resolving such scenarios, providing a structured method to find solutions to limits involving indeterminate forms.