Problem 275 - Calculus Volume 1

Question

For the following exercises, find the horizontal and vertical asymptotes. $$ f(x)=\sin (x) \sin (2 x) $$

Step-by-Step Solution

Answer

There are no horizontal or vertical asymptotes for \( f(x) = \sin(x) \sin(2x) \).

Step 1: Understand the function

The given function is \( f(x) = \sin(x) \sin(2x) \). This is a combination of trigonometric functions. We need to determine both horizontal and vertical asymptotes for this function.

Step 2: Determine vertical asymptotes

Vertical asymptotes occur when the function is undefined or tends to infinity. Since \( \sin(x) \) and \( \sin(2x) \) are trigonometric functions that are defined and finite for all \( x \), there are no vertical asymptotes for the function \( f(x) = \sin(x) \sin(2x) \).

Step 3: Determine horizontal asymptotes

Horizontal asymptotes occur as \( x \to \pm\infty \). The value each \( \sin(x) \) term takes is periodically bounded between -1 and 1. Therefore, as \( x \to \pm\infty \), \( \sin(x) \sin(2x) \) doesn't settle to a specific horizontal value and simply oscillates. Thus, there are no horizontal asymptotes for this function.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental mathematical functions often associated with angles. They are used in a variety of applications such as modeling waves, cycles, and periodic behavior. The most common trigonometric functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). These functions have certain properties:

• They are periodic, meaning they repeat their values in regular intervals.
• The sine and cosine functions have a range from -1 to 1.
• They are defined for all real numbers, but the tangent function has vertical asymptotes where cosine equals zero.

When dealing with compositions like \( f(x) = \sin(x) \sin(2x) \), both factors are trigonometric functions. This means that the resulting function inherits the periodic properties and oscillatory behavior from sine. This is crucial for understanding its asymptotes.

Horizontal Asymptotes

Horizontal asymptotes are lines that a function approaches as the input moves towards positive or negative infinity. They indicate the end behavior of the function.

• If a function approaches a constant value as \( x \to \pm \infty \), that constant value is the horizontal asymptote.
• For rational functions of the form \( f(x) = \frac{P(x)}{Q(x)} \), horizontal asymptotes are determined by the degree of the polynomials in the numerator and denominator.
• However, trigonometric functions like \( \sin(x) \) have no horizontal asymptotes since they are periodic and don't converge to a particular value.

In our case with \( f(x) = \sin(x) \sin(2x) \), as \( x \to \pm \infty \), the function does not settle to a constant because the sine terms continuously oscillate. Therefore, it has no horizontal asymptotes.

Vertical Asymptotes

Vertical asymptotes occur where a function approaches infinity and are typically seen in rational and some trigonometric functions. They indicate points of discontinuity in the domain:

• For rational functions, vertical asymptotes are present where the denominator equals zero, assuming the numerator is not zero at these points.
• In the function \( f(x) = \tan(x) \), vertical asymptotes occur where \( \cos(x) = 0 \), because \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
• However, functions composed solely of sine or cosine do not have vertical asymptotes because they are both defined for all real numbers.

For our specific function \( f(x) = \sin(x) \sin(2x) \), neither \( \sin(x) \) nor \( \sin(2x) \) have conditions that lead to undefined behavior. As a result, the function lacks vertical asymptotes.