Question

In Exercises 47 and 48 , determine whether the graph of the function has a tangent at the origin. Explain your answer. $$f(x)=\left\\{\begin{array}{ll}{x \sin \frac{1}{x},} & {x \neq 0} \\ {0,} & {x=0}\end{array}\right.$$

Short answer

No. The graph of the function does not have a tangent at the origin, as the limit of the derivative of the function \(f(x) = x \sin \frac{1}{x}\) as \(x\) approaches zero does not exist.

Step-by-step solution

Find the derivative of the function for \(x \neq 0\)

The first step involves calculating the derivative of the function \(f(x) = x \sin \frac{1}{x}\) for \(x \neq 0\). The product rule of differentiation is applied, which results in the following: \(f'(x) = \sin \frac{1}{x} - \frac{x \cos \frac{1}{x}}{x^2}\) when \(x \neq 0\).

Evaluate the limit of the derivative as \(x\) approaches zero

In this step, we utilize L'Hopital's rule in order to evaluate \(\lim _{x \to 0} f'(x)\). This involves repeatedly applying the rule until we have a limit that can be readily calculated: \(\lim_{x \to 0} f'(x) = \lim_{x \to 0} (\sin \frac{1}{x} - \frac{x \cos \frac{1}{x}}{x^2}) = 0 - \lim_{x \to 0} \frac{\cos \frac{1}{x}}{2x}\). The limit of the \(\frac{\cos u}{u}\)-term is not determinable; therefore, the limit of the derivative as \(x\) approaches zero does not exist.

Analyze the result

Now we will analyze the result. The derivative of the function at the origin, i.e., the slope of the tangent at the origin, does not exist; therefore, the graph of the function does not have a tangent at the origin.

Key concepts

Product Rule of Differentiation

Understanding the product rule is crucial when dealing with derivatives of functions that are the products of two or more other functions. It's a tool for finding the rate at which such a function is changing at any point. In essence, the product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

For functions denoted as \( u \) and \( v \), the product rule is formally given by \( (uv)' = u'v + uv' \). In the context of our exercise, the function \( f(x) = x \sin \frac{1}{x} \), for \( x eq 0 \), is the product of \( u(x) = x \) and \( v(x) = \sin \frac{1}{x} \). Applying the product rule, we then calculate \( f'(x) \) as shown in the step-by-step solution, showcasing the practical use of this concept in calculus.

L'Hopital's Rule

L'Hopital's rule is a powerful technique to evaluate limits that yield indeterminate forms, such as \( 0/0 \) or \( \infty/\infty \). This rule allows us to find the limit of a ratio of two functions by instead finding the limit of the ratio of their derivatives, provided certain conditions are met.

In our textbook example, L'Hopital's rule was invoked to assess the limit of the derivative as \( x \) approaches zero. However, difficulties arose because the resulting limit led to an indeterminate form that is not easily resolved even with L'Hopital's rule. As a critical tool in calculus, L'Hopital's rule simplifies the process of finding limits but reminds us to be cautious of its applicability when confronted with more complex scenarios.

Limit of a Function

In calculus, the limit of a function is a fundamental concept that describes the behavior of that function as it approaches a particular value or point. Limits help in understanding continuity, derivatives, and the behavior of functions near points of interest. They are pivotal in defining certain functions at points where they would otherwise be undefined.

In our example, we focus on the limit of the derivative \( f'(x) \) as \( x \) approaches zero. Calculating this limit is crucial to determine if there is a tangent at the origin. However, as indicated by the step-by-step solution, the limit of the derivative does not exist in this case. This tells us something important about the nature of the function at that point: the graph of the function \( f(x) \) does not have a tangent line at the origin, suggesting a sort of corner or cusp at that location.

Derivative of a Function

The derivative of a function represents the rate at which the function value changes with respect to a change in its input value. In simpler terms, it measures how steep the curve of the function is at any given point. Derivatives are a cornerstone of differential calculus and are used in various fields, including physics, engineering, and economics, to model changes and understand the dynamism of systems.

In our exercise, finding the derivative of \( f(x) \) for \( x eq 0 \) is the first step to determine if a tangent exists at the origin. When the outcome of this derivative evaluation at \( x = 0 \) leads to a non-existent limit, as pointed out in the steps provided, it implies that the slope of the tangent line (if one existed) is not defined. This lack of a defined derivative at the origin confirms that the graph of the function fails to have a tangent line there, emphasizing the importance of derivatives in analyzing the geometry of function graphs.