Problem 100 - Calculus of a Single Variable

Question

Writing Use a graphing utility to graph $$ f(x)=x, \quad g(x)=\sin ^{2} x, \quad \text { and } \quad h(x)=\frac{\sin ^{2} x}{x} $$ in the same viewing window. Compare the magnitudes of \(f(x)\) and \(g(x)\) when \(x\) is close to \(0 .\) Use the comparison to write a short paragraph explaining why \(\lim _{x \rightarrow 0} h(x)=0\)

Step-by-Step Solution

Answer

Graphing the functions \(f(x)=x\), \(g(x)=\sin^2 x\), and \(h(x)=\frac{\sin^2 x}{x}\) allows us to visualize their behavior at \(x\) near \(0\). It is observed that the magnitude of \(g(x)=\sin^2 x\) is less than or equal to that of \(f(x)=x\) near \(0\). Given that \(h(x)\) is the ratio of \(g(x)\) to \(f(x)\), it follows that \(h(x)\) decreases to \(0\) as \(x\) approaches \(0\). Therefore, \(\lim _{x \rightarrow 0} h(x) = 0\).

Step 1: Graph the Functions

First, use your graphing utility to plot each of the three functions \(f(x) = x\), \(g(x) = \sin^2 x\), and \(h(x) = \frac{\sin^2 x}{x}\) in the same viewing window. Remember that \(\sin^2 x\) refers to the square of the sine of \(x\). This will give you a visual representation of how these functions behave.

Step 2: Compare the Magnitudes of f(x) and g(x) near 0

After graphing, look at the area of the graph near where \(x\) is close to \(0\). Focus on the functions \(f(x)\) and \(g(x)\). Particularly, the key point to observe is that as \(x\) approaches \(0\), the magnitude of \(g(x)=\sin^2 x\) is smaller or equal to that of \(f(x)=x\). This is true because the maximum value of \(\sin^2 x\) is \(1\), and \(0<=|\sin^2 x|<=|x|\) whenever \(|x|>=1\).

Step 3: Analyze the Limit of h(x) as x approaches 0

With the magnitudes of \(f(x)\) and \(g(x)\) determined close to \(0\), you can infer the behavior of \(h(x)\). Specifically, \(h(x)\) is equal to \(\frac{g(x)}{f(x)}\). Since \(|g(x)|<=|f(x)|\) near \(0\), and considering that any nonzero number divided by itself gives \(1\) while a number less than itself gives a fraction less than \(1\), it follows that \(h(x)\) approaches \(0\) as \(x\) approaches \(0\). Hence, it can be stated that \(\lim _{x \rightarrow 0} h(x) = 0\).

Key Concepts

Graphing Functions

Graphing functions is an essential skill in calculus to visualize how a function behaves over a particular domain. To graph the functions \( f(x) = x \), \( g(x) = \sin^2 x \), and \( h(x) = \frac{\sin^2 x}{x} \), you'll want to use a graphing utility or calculator capable of plotting multiple functions simultaneously.
This helps to examine how each function behaves as x changes, especially around critical points like zero. When graphing these functions:

• \( f(x) = x \): This is a straight line passing through the origin with a slope of 1. It's straightforward, linear.
• \( g(x) = \sin^2 x \): The sine function, squared. This creates a wave that fluctuates between 0 and 1.
• \( h(x) = \frac{\sin^2 x}{x} \): This function is more complex. Near zero, it requires special attention due to being undefined at \( x = 0 \).

When these functions are plotted in the same window, their interactions and intersections provide insights into their comparative behaviors as x approaches zero.

Limit Laws

Limit laws are fundamental rules used to calculate limits in calculus. They help us understand how functions behave as they approach specific values. When applying limit laws to the functions in question, we focus on \( h(x) = \frac{\sin^2 x}{x} \) as \( x \) approaches zero.
Why do we use limit laws here?

• The Limit of a Quotient Law states that the limit of a quotient is the quotient of the limits, given they exist.
• Limit laws also assert that if a function can be squeezed between two functions approaching the same limit, the original function will share this limit.

Thus, since \( \sin^2 x \) is bounded and \( \lim_{x \to 0} x = 0 \), the ratio \( \frac{ \sin^2 x }{ x } \) approaches 0. This is a direct application of these limit principles, showing effectively that such limit laws make analyzing boundaries and behaviors simpler and more intuitive.

Sine Function

The sine function is a cornerstone of trigonometry and calculus. The function \( g(x) = \sin^2 x \) is derived from this fundamental wave function, \( \sin x \).
Here's what you need to know about sine functions:

• The basic sine function, \( \sin x \), oscillates between -1 and 1.
• \( \sin^2 x \), being squared, ensures that the range is between 0 and 1, providing a non-negative value.
• This transformation flattens out the wave into upward arcs, critical for analyzing \( g(x) \) and integral to understanding the limit \( h(x) = \frac{\sin^2 x}{x} \).

Understanding these properties allows you to appreciate how functions like \( h(x) \) exhibit their peculiar behaviors. The sine function's natural cycle influences both the function's slope and its limit characteristics near zero.

Behavior of Functions near Zero

Analysing how functions behave as they approach zero reveals crucial insights into their limits and continuity. For \( h(x) = \frac{\sin^2 x}{x} \) as \( x \) nears zero, special attention is required.
Some points to consider:

• As \( x \) approaches zero, \( g(x) = \sin^2 x \) oscillates but remains small, generally moving towards zero. This is because \( \sin(0) = 0 \).
• \( f(x) = x \) straightforwardly approaches zero because it is directly dependent on \( x \).
• For \( h(x) \), the ratio format \( \frac{\sin^2 x}{x} \) indicates that it is undefined at \( x = 0 \), but due to the small values of \( \sin^2 x \), the function approaches zero.

Notice the behavior around zero. Limits and continuity principles allow us to make conclusions about functions even where direct substitution fails. This evaluation shows why the function values are pushed towards zero, explaining \( \lim_{x \to 0} h(x) = 0 \).