Problem 99 - Calculus of a Single Variable

Question

Writing Use a graphing utility to graph $$ f(x)=x, g(x)=\sin x, \quad \text { and } \quad h(x)=\frac{\sin x}{x} $$ 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 \(\lim _{x \rightarrow 0} h(x)=1\)

Step-by-Step Solution

Answer

Upon graphing and comparing the functions \(f(x)\) and \(g(x)\) as \(x\) approaches zero, it is observed that their values get very close. This indicates that for small values of \(x\), \(\sin x \approx x\). This observation is used to analyse \(h(x) = \frac{\sin x}{x}\), asserting that as \(x\) approaches zero, this function becomes \(1\). Thus, \(\lim _{x \rightarrow 0} h(x) = 1\).

Step 1: Graph the functions

First, graph the functions \(f(x)=x\), \(g(x)=\sin x\), and \(h(x)=\frac{\sin x}{x}\) using a graphing utility. Ensure all three functions are on the same viewing window for a clear comparison.

Step 2: Compare f(x) and g(x) at \(x = 0\)

Analyze the graph and observe the behaviour of \(f(x)\) and \(g(x)\) when \(x\) is close to zero. Notice that the value of \(f(x)\) and \(g(x)\) are very close to each other in this range. In fact, as \(x\) approaches zero, the value of \(\sin x\) (which is \(g(x)\) ) approaches \(x\), this is because for small values of \(x\), \(\sin x \approx x\).

Step 3: Analyse h(x) and confirm the limit

Looking at \(h(x) = \frac{\sin x}{x}\), it can be seen that as \(x\) approaches zero, this function becomes \(\frac{\sin x}{x}\) or essentially \(\frac{x}{x}\), which is 1. Thus, we can conclude that \(\lim _{x \rightarrow 0} h(x) = 1\).

Key Concepts

Graphing functions

Graphing functions is a fundamental skill in understanding calculus and analyzing the behavior of functions. When you graph functions such as \(f(x) = x\), \(g(x) = \sin x\), and \(h(x) = \frac{\sin x}{x}\), you are visualizing how these mathematical expressions behave relative to each other across a range of \(x\) values. Using a graphing utility like Desmos, GeoGebra, or a calculator can help you see how these functions interact in one viewing window.

Here’s a quick checklist on why graphing is beneficial:

• It provides a visual understanding of function behavior and relationships.
• It helps identify key points, such as intersections and asymptotes.
• It serves as a supportive tool for solving limits and analyzing function continuity.

When graphed together, you may notice that near \(x = 0\), these functions reveal interesting comparisons. This visual comparison is crucial for understanding limits and function behavior.

Limit comparison

Limit comparison involves examining the behavior of functions as \(x\) approaches a particular value, often to determine limits or similarities between functions. This approach is particularly useful when looking at \(f(x) = x\) and \(g(x) = \sin x\) for values close to zero. As \(x\) approaches zero, it is important to witness how \(\sin x\) appears almost identical to \(x\). This similarity forms the basis of limit comparison in this scenario.

You can use the following insights when approaching limit comparisons:

• When two functions behave similarly in an interval, their limits as \(x\) approaches a given point can be compared directly.
• If \(\sin x \approx x\) for small \(x\), then generally, their ratios also approach a specific value.
• Limit comparison helps establish or confirm that \(\lim_{x \to 0} \frac{\sin x}{x} = 1\).

Understanding these behaviors through graphs and calculations solidifies your grasp of limit concepts, providing evidence for intuitive conclusions.

Behavior of sine function near zero

The behavior of the sine function near zero is a cornerstone concept in calculus and trigonometry. For small angles, the sine of an angle \(x\) (where \(x\) is in radians) behaves approximately like the angle itself. This is captured by the small-angle approximation: \(\sin x \approx x\) when \(x\) is around zero.

Here are the key points to remember regarding this approximation:

• When \(x\) is close to zero, \(\sin x\) converges to \(x\) because the differences between the two become negligible.
• This behavior underlies many derivations and proofs in calculus, notably the limit \(\lim_{x \to 0} \frac{\sin x}{x} = 1\).
• It is a critical concept when dealing with oscillatory behaviors and wave functions.

This concept simplifies complex mathematical problems by making it easier to work with trigonometric identities and limits, particularly in calculus while fostering a deeper understanding of function behaviors.