Question

Use a calculator to graph the function and estimate the value of the limit, then use L'Hôpital's rule to find the limit directly. $$ \lim _{x \rightarrow 0} x \sin \left(\frac{1}{x}\right) $$

Short answer

The limit is 0 as \( x \to 0 \).

Step-by-step solution

Understanding the Function

The function we are considering is \( x \sin\left(\frac{1}{x}\right) \). As \( x \) approaches zero, the argument of the sine function, \( \frac{1}{x} \), approaches infinity. This causes the sine function to oscillate indefinitely between -1 and 1.

Graphing the Function

Use a graphing calculator to plot the function \( x \sin\left(\frac{1}{x}\right) \) for values of \( x \) around zero (e.g., from -0.1 to 0.1). Observe the behavior of the graph near \( x = 0 \). You'll notice the graph oscillates rapidly and becomes confined closer to the x-axis as it approaches zero, suggesting the limit might be 0.

Applying L'Hôpital's Rule Setup

Verify that L'Hôpital's Rule is applicable. The form \( \lim _{x \rightarrow 0} x \sin\left(\frac{1}{x}\right) \) is of the indeterminate form \( 0 \times \pm 1 \). We rewrite this as \[ \lim _{x \rightarrow 0} \frac{\sin\left(\frac{1}{x}\right)}{\frac{1}{x}} \] to clarify its indeterminate form \( \frac{0}{\infty} \).

L'Hôpital's Rule Application

The expression \( \frac{\sin\left(\frac{1}{x}\right)}{\frac{1}{x}} \) can also be written as \( \frac{\sin(y)}{y} \) by substituting \( y = \frac{1}{x} \), where \( y \rightarrow \infty \) as \( x \rightarrow 0 \). The limit of \( \frac{\sin(y)}{y} \) as \( y \rightarrow \infty \) does not approach a determinate value but oscillates. Thus applying L'Hôpital’s Rule is unconventional here; instead, realize L'Hôpital’s standard form does not strictly apply.

Estimating Limit by Bound Comparison

Even if L'Hôpital’s Rule doesn't directly apply, evaluate \( -|x| \leq x \sin\left(\frac{1}{x}\right) \leq |x| \). As \( x \to 0 \), both \( -|x| \) and \( |x| \to 0 \), showing \( x \sin\left(\frac{1}{x}\right) \to 0 \) by the Squeeze Theorem.

Key concepts

Calculus Limits

Limits in calculus describe the behavior of a function as it approaches a particular point. In mathematical notation, the limit of a function \( f(x) \) as \( x \) approaches a value \( a \) is written as \( \lim_{x \to a} f(x) \). This evaluates how \( f(x) \) behaves as \( x \) gets closer to \( a \), but not necessarily reaching \( a \) itself.

When considering limits, we evaluate whether a function approaches a specific value or meaningfully describes the function's behavior despite seeming contradictions. For our function, \( x \sin\left(\frac{1}{x}\right) \), as \( x \to 0 \), manually checking very small values of \( x \) using a calculator or graph can provide insights into the limit's possible value. Observing that the graph persuades visualization of patterns, even in such erratic oscillations, is necessary for calculus students.

Recognizing rapid oscillations around zero can guide the expectation that \( \lim_{x \to 0} x \sin\left(\frac{1}{x}\right) = 0 \). This arises because the sine function causes values to swing between -1 and 1, shrinking them multiplicatively by \( x \)'s small scale.

Squeeze Theorem

The Squeeze Theorem plays a crucial role in determining limits of functions that oscillate between two boundary lines. It helps when direct approaches to limits are challenging or when oscillations are involved. The theorem states if a function \( g(x) \) is "squeezed" between two other functions \( f(x) \) and \( h(x) \), and if both \( f(x) \) and \( h(x) \) approach the same limit as \( x \to a \), then \( g(x) \) also approaches that limit.

For example, in the exercise, the function \( x \sin\left(\frac{1}{x}\right) \) is squeezed by bounds \(-|x|\) and \(|x|\). This is summarized by:

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

As \( x \to 0 \), both \(-|x|\) and \(|x|\) converge to zero, thus squeezing \( x \sin\left(\frac{1}{x}\right) \) to zero too. Such reasoning is vital when analyzing limits where the function does not naturally simplify or stabilize.

Indeterminate Forms

Indeterminate forms arise in calculus when expressions fail to provide definitive information about a limit. They occur when substitution leads to results like \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 0 \times \infty \), and others.

For the expression \( x \sin\left(\frac{1}{x}\right) \), attempting to substitute \( x = 0 \) yields the form \( 0 \times \pm 1 \). This form typically doesn't convey a straightforward answer. We must interpret them with tools like L'Hôpital's Rule or the Squeeze Theorem.

L'Hôpital's Rule helps find limits for indeterminate forms by deriving functions. For our expression, rearranging into \( \frac{\sin\left(\frac{1}{x}\right)}{\frac{1}{x}} \) might suggest applying L'Hôpital's, but with an unconventional approach being less helpful here—since \( \sin(y) \) remains undefined at extremes, the Squeeze Theorem prevails as an effective strategy. Understanding these forms prompts deeper scrutiny and appropriate methods for limit resolution.