Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

Plot the functions \(u(x), l(x)\), and \(f(x)\). Then use these graphs along with the Squeeze Theorem to determine \(\lim _{x \rightarrow 0} f(x)\). $$ u(x)=1, l(x)=1-x^{2}, f(x)=\cos ^{2} x $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify the Functions

We are given three functions: \( u(x) = 1 \), \( l(x) = 1 - x^2 \), and \( f(x) = \cos^2 x \). Our goal is to plot these functions over an interval that helps analyze the behavior as \( x \to 0 \).

Step 2: Understand the Interval

Since we want to find the limit as \( x \to 0 \), it's beneficial to consider a small interval around zero, such as \(-1 \leq x \leq 1\), to capture the relevant behavior of \( f(x) \) with respect to \( u(x) \) and \( l(x) \).

Step 3: Plot the Functions

Let's plot \( u(x) = 1 \) as a horizontal line, \( l(x) = 1 - x^2 \) which is a downward facing parabola opening from \( y = 1 \) at \( x = 0 \), and \( f(x) = \cos^2 x \), which oscillates between \( 0 \) and \( 1 \) with \( f(0) = 1 \). All graphs are considered within the interval \(-1 \leq x \leq 1\).

Step 4: Analyze Graphs and Apply the Squeeze Theorem

The Squeeze Theorem states that if \( l(x) \leq f(x) \leq u(x) \) near \( x = 0 \) and \( \lim_{x \to 0} l(x) = \lim_{x \to 0} u(x) = L \), then \( \lim_{x \to 0} f(x) = L \). Here, \( l(x) = 1 - x^2 \to 1 \) and \( u(x) = 1 \to 1 \) as \( x \to 0 \). Also, \( 1 - x^2 \leq \cos^2 x \leq 1 \). Therefore, \( \lim_{x \to 0} f(x) = 1 \).

Step 5: Conclusion

By the graph and the Squeeze Theorem, we determine that as \( x \approaches 0 \), \( f(x) = \cos^2 x \) is squeezed to the limit \( 1 \). Thus, the limit is \( 1 \).

Key Concepts

Functions Plotting

Plotting functions is an essential early step in understanding and analyzing mathematical behaviors. It allows us to visualize and compare the behavior of different functions over a specific interval. For the given exercise, we have three functions:

• \( u(x) = 1 \): This is a constant function, resulting in a horizontal line at \( y = 1 \) on the graph.
• \( l(x) = 1 - x^2 \): This represents a downward-facing parabola. It's highest at \( x = 0 \) where \( y = 1 \), and it drops off as \( x \) moves away from zero.
• \( f(x) = \cos^2 x \): The cosine function squared, which oscillates between \( 0 \) and \( 1 \).

By plotting these over the interval \(-1 \leq x \leq 1\), we'll observe how each behaves around zero. In this range, the highest, \( u(x) \), stays constant at \( y = 1 \), while \( l(x) \) bows downward to depict the parabolic nature. The function \( f(x) \) oscillates, creating a wave-like pattern, yet always stays between the values of \( l(x) \) and \( u(x) \). This visual setup aids us in understanding how functions interact and helps to apply concepts like the Squeeze Theorem effectively.

Limit Calculation

Limit calculation is a fundamental concept in calculus that helps us understand the behavior of a function as it approaches a specific point. In this exercise, we are interested in the behavior of \( f(x) = \cos^2 x \) as \( x \rightarrow 0 \). The Squeeze Theorem, also known as the Sandwich Theorem, is particularly useful in determining this limit when direct substitution isn't clear.
To apply the theorem:

• Identify three functions: a lower bound \( l(x) = 1 - x^2 \), an upper bound \( u(x) = 1 \), and the target function \( f(x) = \cos^2 x \) we aim to find the limit for.
• For \( x \) near 0, ensure that \( l(x) \leq f(x) \leq u(x) \). Over the interval chosen, this holds true, as noted from their plotted graphs.
• Evaluate the limit of \( l(x) \) and \( u(x) \) as \( x \rightarrow 0 \). Both \( l(x) \) and \( u(x) \) simplify to \( 1 \), meeting the theorem's conditions.

Thus, using the Squeeze Theorem, we conclude that the limit of \( f(x) \) as \( x \rightarrow 0 \) must also be \( 1 \). This process allows us to mathematically confirm what we're observing from the model and solidifies our understanding of the function's behavior at that point.

Cosine Function

The cosine function, denoted as \( \cos(x) \), is a fundamental trigonometric function that is periodic and oscillates between -1 and 1. When squared, as in \( f(x) = \cos^2 x \), the oscillation remains but shifts between 0 and 1 instead, providing a different range of values for its outputs.
A few key properties of the cosine function and its square are:

• The function is even, meaning \( \cos(-x) = \cos(x) \), indicating symmetry about the y-axis.
• \( \cos^2 x \) allows periodicity with a period of \( \pi \), due to the nature of the squared cosine cycle.
• At \( x = 0 \), \( \cos^2 0 = 1^2 = 1 \), which is relevant for limit calculation as \( x \rightarrow 0 \).

In the context of the squeeze theorem, the inherent properties of \( \cos^2 x \), like its bounded range, provide an excellent scenario where the theorem applies cleanly. Recognizing this behavior assists in evaluating limits and further understanding trigonometric functions in calculus.