Question

Using the Squeeze Theorem In Exercises \(91-94\) , use a graphing utility to graph the given function and the equations \(y=|x|\) and \(y=-|x|\) in the same viewing window. Using the graphs to observe the Squeeze Theorem visually, find \(\lim _{x \rightarrow 0} f(x)\) $$ h(x)=x \cos \frac{1}{x} $$

Short answer

By directly applying the Squeeze Theorem, the limit of the function \(h(x) = x \cos \frac{1}{x}\) as \(x\) approaches 0 is 0.

Step-by-step solution

Understand the Function

Given the function \(h(x) = x \cos \frac{1}{x}\). This function appears to oscillate between positive and negative values when approaching the x-value of 0.

Plot the Function and bounding functions

Using a graphing utility, plot the function \(h(x) = x \cos \frac{1}{x}\) alongside the functions \(y=|x|\) and \(y=-|x|\). You will see that as \(x\) approaches 0, the function \(h(x)\) is squeezed between the other two functions.

Apply the Squeeze Theorem

Since \(h(x)\) is squeezed between \(y=-|x|\) and \(y=|x|\), as \(x\) approaches 0, both \(y=-|x|\) and \(y=|x|\) tend towards 0. Therefore, according to the Squeeze theorem, \(\lim _{x \rightarrow 0} h(x)\) must also be equal to 0.

Key concepts

Limit of a function

Understanding the limit of a function is a fundamental concept in calculus. When we talk about the limit of a function as it approaches a particular point, we are interested in what value the function gets closer to as the input approaches that point. In mathematical terms, we write this as \( \lim_{x \to a} f(x) = L \), which means that as \( x \) approaches \( a \), the values of \( f(x) \) get arbitrarily close to \( L \).
In the context of the Squeeze Theorem, evaluating limits becomes a strategic process. The given exercise involves using this theorem to find the limit of \( h(x) = x \cos \frac{1}{x} \) as \( x \) approaches 0. Knowing that the function oscillates, the Squeeze Theorem allows us to sandwich \( h(x) \) between two simpler functions whose limits we can easily find, thereby determining the limit of \( h(x) \) itself.

• The Squeeze Theorem is especially useful when a function is difficult to evaluate directly because it oscillates or behaves unpredictably near the point of interest.
• Both bounding functions, in this case \( y = |x| \) and \( y = -|x| \), converge to the same limit (0) as \( x \to 0 \).

Therefore, the limit of the function \( h(x) \) is proven to also approach 0.

Oscillating functions

Oscillating functions are functions that exhibit repeated variations, typically switching back-and-forth between certain values. This behavior can make evaluating limits particularly tricky. The function described in this exercise, \( h(x) = x \cos \frac{1}{x} \), is an oscillating function; it varies rapidly as \( x \) nears 0 due to the cosine term \( \cos \frac{1}{x} \).
This rapid oscillation doesn't prevent us from evaluating the function's limit, thanks to the Squeeze Theorem. Even if \( h(x) \) oscillates widely, if we can find two other simpler functions that it gets "squeezed" between, we can determine its limiting behavior at a point. In this case:

• The oscillation comes from the cosine term, which remains bounded between -1 and 1 regardless of the value of \( x \).
• When multiplied by \( x \), the result is a function that remains confined within the bounds set by \( y = |x| \) and \( y = -|x| \).

Recognizing the bounding behavior gives us the power to apply the Squeeze Theorem effectively to find the limit.

Graphical representation of limits

Visualizing limits through graphical representations can make the process of understanding them much easier. By using a graph, we can see how as \( x \) approaches a particular value, the function behaves and where it seems to be heading.
In the case of \( h(x) = x \cos \frac{1}{x} \), graphing helps us observe the function's oscillation and bounding. When plotted along with \( y = |x| \) and \( y = -|x| \), we see that \( h(x) \) remains confined within the two lines as \( x \to 0 \).

• Graphing reveals that the oscillations occur within a narrowing band defined by the bounding functions.
• This diminishing band visually confirms that the limit of \( h(x) \) as \( x \to 0 \) must be 0.

The visual evidence makes it much more intuitive to accept the conclusion provided by the Squeeze Theorem. Looking at a graph, we gain a direct understanding of how the function behaves near 0.