Problem 9 - Thomas Calculus

Question

Find the limits in Exercises \(9-12\) . $$\lim _{x \rightarrow \infty} \frac{\sin 2 x}{x}$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Analyze the Limit Expression

We are tasked to find the limit as \( x \) approaches infinity for the function \( \frac{\sin 2x}{x} \). First, note that \( \sin 2x \) oscillates between -1 and 1 for all values of \( x \). This means that \( -1 \leq \sin 2x \leq 1 \).

Step 2: Apply the Squeeze Theorem

Given that \( -1 \leq \sin 2x \leq 1 \), we can divide the entire inequality by \( x \) (where \( x > 0 \)) to get \( -\frac{1}{x} \leq \frac{\sin 2x}{x} \leq \frac{1}{x} \).

Step 3: Evaluate Limits of Bounds

Consider the limits of the bounds as \( x \rightarrow \infty \): - \( \lim_{x \to \infty} \left(-\frac{1}{x}\right) = 0 \) - \( \lim_{x \to \infty} \left(\frac{1}{x}\right) = 0 \)

Step 4: Conclude Using the Squeeze Theorem

Since \( -\frac{1}{x} \leq \frac{\sin 2x}{x} \leq \frac{1}{x} \) and both bounds approach 0 as \( x \rightarrow \infty \), by the Squeeze Theorem, the limit \( \lim_{x \to \infty} \frac{\sin 2x}{x} \) must also be 0.

Key Concepts

Squeeze Theorem

The Squeeze Theorem is a powerful tool in calculus. It helps us find the limit of a function when the direct evaluation might be challenging.
To use the Squeeze Theorem, we need three functions: two that "squeeze" the function of interest.
Essentially, if we have a function, say \( f(x) \), and we know that it is always greater than another function \( g(x) \) and less than \( h(x) \), and both \( g(x) \) and \( h(x) \) approach the same limit as \( x \) approaches a particular value, then \( f(x) \) must approach that same limit.

• If \( g(x) \leq f(x) \leq h(x) \) for all \( x \), and both \( \lim_{x \to a} g(x) = L \) and \( \lim_{x \to a} h(x) = L \), then \( \lim_{x \to a} f(x) = L \) as well.

In our example, the function \( \frac{\sin 2x}{x} \) is squeezed between \( -\frac{1}{x} \) and \( \frac{1}{x} \). As both these functions approach zero as \( x \) goes to infinity, so does \( \frac{\sin 2x}{x} \).
This makes the Squeeze Theorem a very effective method for proving the limits of functions that might be sandwiched between two simpler limits.

Infinity Limits

Infinity limits deal with the behavior of functions as the input, often denoted as \( x \), grows larger without bound or approaches negative infinity. These limits help us understand how functions behave in extreme cases.
Calculating limits as \( x \to \infty \) can help in predicting long-term behavior in models and scenarios such as population growth or economic trends.

• These limits are common in many mathematical models where variables increase without end.
• When evaluating infinity limits, look to simplify functions and expressions using known limits or approximations.

In our particular exercise, we're considering \( \lim_{x \to \infty} \frac{\sin 2x}{x} \).
Here, \( \sin 2x \) keeps oscillating between 1 and -1 no matter how big \( x \) gets, but the denominator \( x \) continues to grow. Therefore, our function's overall magnitude shrinks, tending to zero as \( x \) becomes very large.

Trigonometric Functions

Trigonometric functions are fundamental in both pure and applied mathematics. They describe relationships in triangles and model periodic phenomena such as sound and light waves.
Key trigonometric functions include sine, cosine, and tangent, which relate the angles of a right triangle to the ratios of its sides.
The sine function, represented as \( \sin x \), has values ranging between -1 and 1.
It is periodic with a period of \( 2\pi \), meaning the function repeats its values in every interval of \( 2\pi \).

• In calculus, \( \sin x \) and other trigonometric functions often appear in limits and integrals.
• Their periodic nature makes them interesting for limit problems, especially when combined with non-periodic functions like polynomials.

In the example, \( \sin 2x \) oscillates between -1 and 1, but when divided by \( x \), it illustrates how its influence diminishes as \( x \) increases. Understanding this diminishing effect is crucial for evaluating the limit as \( x \to \infty \).