Problem 276 - Calculus Volume 1

Question

Find the horizontal and vertical asymptotes. \(f(x)=\cos x+\cos (3 x)+\cos (5 x)\)

Step-by-Step Solution

Answer

The function has no vertical or horizontal asymptotes.

Step 1: Understand the Type of Function

The function provided is a trigonometric function, specifically a sum of cosine functions. Trigonometric functions often involve sinusoidal behavior on all real numbers, and they generally do not have asymptotes unless transformed.

Step 2: Analyze Vertical Asymptotes

Vertical asymptotes occur in functions where the output becomes infinitely large or small at certain input values. Trigonometric functions like cosine do not have vertical asymptotes because cosine is defined and finite for all real numbers. Therefore, the function \(f(x) = \cos x + \cos (3x) + \cos (5x)\) does not have any vertical asymptotes.

Step 3: Analyze Horizontal Asymptotes

Horizontal asymptotes are limits reached as x approaches infinity or negative infinity. Since cosine functions oscillate between -1 and 1, and the sum of a finite number of such functions remains bounded, \(f(x)\) will not approach a single value as \(x\) tends to infinity. Therefore, \(f(x) = \cos x + \cos (3x) + \cos (5x)\) does not have any horizontal asymptotes either.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are lines where a function's values grow infinitely large, either positively or negatively, as the input approaches a specific point. They occur when a function is undefined for some input values. For instance, in a rational function, vertical asymptotes often occur where the denominator is zero. However, not all functions have vertical asymptotes.
In trigonometric functions like \((x) = \cos x + \cos(3x) + \cos(5x)\), vertical asymptotes are not present. Trigonometric functions such as cosine are defined for all real numbers, \(-\infty < x < +\infty\). This means there's no point where the function becomes undefined, and hence, no vertical asymptotes to be found. This property makes trigonometric functions quite different from some algebraic functions in terms of their continuity and behavior over the real number line.

Horizontal Asymptotes

Horizontal asymptotes refer to a line that a function approaches as the input becomes very large (positive or negative infinity). These lines signify the values a function approaches, but never quite reaches. In rational functions, a horizontal asymptote can indicate long-term behavior. However, the case is different for trigonometric functions.
Cosine functions oscillate between -1 and 1 as their input values change. This oscillation is characteristic of cosine due to its periodic nature. Therefore, when we consider a function like \(f(x) = \cos x + \cos(3x) + \cos(5x)\), the function is always bounded between certain values and never settles at a single solution as \(x\) approaches infinity or negative infinity. Consequently, such a function does not exhibit horizontal asymptotes, because there is no single horizontal value that the function endlessly approaches.

Cosine Function

The cosine function is one of the primary trigonometric functions, known for its wave-like behavior and periodic nature. Represented as \(cos(x)\), this function repeats its pattern every \(2\pi\) units, which makes it periodic. It is defined for all real numbers and produces values ranging from -1 to 1.

• Cosine is an even function, which means \(cos(-x) = cos(x)\).
• Its graph is a smooth wave that cycles every \(2\pi\).
• When combined with functions like \(cos(3x)\) or \(cos(5x)\), the waves interfere, creating complex waveforms.

The cosine function is crucial in various fields such as physics and engineering, especially in describing phenomena like sound and light waves. When you sum multiple cosine functions, you see how these simple waveforms can add up to produce intricate patterns without resulting in asymptotes.