Problem 65 - Algebra and Trigonometry

Question

Graph the function \(f(x)=\frac{\sin x}{x}, x>0 .\) Based on the graph what do you conjecture about the value of \(\frac{\sin x}{x}\) for \(x\) close to \(0 ?\)

Step-by-Step Solution

Answer

As \(x\) approaches 0, \(\frac{\sin x}{x}\) approaches 1.

Step 1 - Identify the function

The given function is \(f(x) = \frac{\sin x}{x}\) for \(x > 0\).

Step 2 - Consider the behavior as \(x\) approaches 0

To better understand the behavior of \(f(x)\) as \(x\) gets close to zero, it is helpful to graph the function \(f(x) = \frac{\sin x}{x}\) for values of \(x\) near zero but greater than zero.

Step 3 - Use a graphing tool

Plot the function \(f(x) = \frac{\sin x}{x}\) using a graphing tool or software. Make sure to choose a range of \(x\) values close to 0, such as from 0.01 to 1.

Step 4 - Observe the graph

Observe the shape of the graph as \(x\) approaches 0 from the right. Notice that the graph appears to approach a specific value as \(x\) gets closer to 0.

Step 5 - Conjecture the limiting value

Based on the graph, it can be observed that as \(x\) approaches 0, \(\frac{\sin x}{x}\) seems to approach the value 1. This suggests that the limit of \(\frac{\sin x}{x}\) as \(x\) approaches 0 from the right is 1.

Key Concepts

Graphing functions

Graphing functions can help us visualize and understand the behavior of mathematical expressions.
To graph the function \(f(x) = \frac{\sin x}{x}\) for \(x > 0\), you can use graphing tools like Desmos or a graphing calculator.
When graphing, it's important to focus on an appropriate range. In this exercise, we are interested in values of \(x\) close to zero.
For example, consider graphing from \0.01\ to \1\. This gives a clearer picture of how the function behaves near zero.
Observing graphs visually represents trends and tendencies, like whether the function rises, falls, or stabilizes.

Behavior near zero

The behavior of a function near zero is crucial in understanding limits and continuity.
When we graph \(\frac{\sin x}{x}\) and look at values close to zero, we can observe how the function behaves.
For \(x > 0\) and approaching zero from the right side, the graph helps illustrate any tendencies.
This behavior can reveal important information about the function's limit as \(x\) nears zero.
By graphing \(f(x) = \frac{\sin x}{x}\) for small values of \(x\), it shows that the function approaches a value of 1.
This suggests a stable behavior near zero, helping us understand its properties better.

Limits

The concept of limits helps determine the value that a function approaches as the input gets close to a certain point.
Here, we're interested in \(\lim_{{x\to 0^+}} \frac{\sin x}{x}\).
Graphing the function \(\frac{\sin x}{x}\) as \(x\) approaches zero from the right shows that the function's value gets closer and closer to 1.
This visual evidence supports the conjecture that the limit is indeed 1.
Using limit notation, we write this as \(\lim_{{x\to 0^+}} \frac{\sin x}{x} = 1\).
The concept of limits is fundamental in calculus, offering insights into the behavior of functions at critical points.