Problem 69 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. For the graph of \(y=2^{x} \sin x,\) as \(x\) approaches \(-\infty\) \(y\) approaches \(0 .\)

Step-by-Step Solution

Answer

The statement is true. As \(x\) approaches \(-\infty\), the term \(2^x \sin{x}\) will approach 0.

Step 1: Analyze the Exponential Part

We first examine the exponential part of the function \(2^x\). It is known that as \(x\) approaches \(-\infty\), \(2^x\) approaches 0, because the the base of the exponential is greater than 1.

Step 2: Analyze the Trigonometric Part

Next, observe the trigonometric part of the function, \(\sin{x}\). The sine function oscillates between -1 and 1 for all real numbers.

Step 3: Combine the behaviors

When we combine these behaviors, we understand that \(2^x \sin x\) is the product of a function that approaches 0 and a function that oscillates between -1 and 1. The 0 will dominate this interaction causing the entire function to approach 0 as \(x\) approaches \(-\infty\).

Key Concepts

Limits and Asymptotic Behavior

In mathematics, understanding the concept of limits and asymptotic behavior is crucial for analyzing how functions behave as they approach certain points, such as infinity. The limit helps us to describe what happens to the function's output (or "y" value) as the input ("x" value) approaches a particular value or infinity.

An asymptote is often a line that the graph of a function approaches but never actually reaches. In the mentioned exercise, the focus is on how the value of the function behaves as the input, or "x", approaches negative infinity. It is common to explore limits as they approach both positive and negative infinity to predict the behavior of graphs.

• Limits provide a way to describe the behavior of a function from a distance or at extreme values.
• Understanding asymptotic behavior allows us to predict how graphs will look beyond the visible range.
• In the exercise, observing the function's limit as \( x \to -\infty \), we conclude that the function \( y = 2^x \sin x \) approaches 0.

Exponential Functions

Exponential functions are powerful mathematical tools characterized by the presence of a constant base raised to a variable exponent, such as \(2^x\). These functions have specific, predictable behaviors, especially as the variable approaches infinity or negative infinity.

Generally, for exponential functions with a base greater than 1, as the variable \(x\) approaches negative infinity, the value of the function approaches zero. This is because the exponent becomes a very large negative number, making the overall value incredibly small.

• In our exercise, as \(x \to -\infty\), \(2^x\) falls towards zero due to its rapidly decreasing positive value.
• This characteristic of exponential growth or decay is frequently used in real-world contexts, such as population growth or radioactive decay.
• Recognizing these patterns helps in predicting the behavior of more complex functions that involve exponentials.

Trigonometric Functions

Trigonometric functions, such as \( \sin x \), are periodic functions that repeat their values in regular intervals, generally described by their wave-like oscillations. For sine and cosine functions, these oscillations occur between the values of -1 and 1.

This periodic nature is what causes trigonometric functions to have predictable oscillations, no matter how large or small \(x\) becomes. They do not converge to a single value as \(x\) approaches infinity or negative infinity but continue oscillating.

• In the exercise, \( \sin x \) maintains its oscillation between -1 and 1 as \(x\) ranges over all real numbers.
• This property is key in determining the overall behavior of the product \(2^x \sin x\).
• Although \( \sin x \) does not approach a limit as \(x\) tends towards negative infinity, it influences but does not dominate the limit behavior of the combined function.