Question

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. $$f(x)=\sin x$$

Short answer

Answer: The sine function oscillates between 1 and -1 and does not have a specific end behavior as $$x \to \pm\infty$$. It does not settle to any particular value and does not have asymptotes.

Step-by-step solution

Understand the sine function

The sine function is a periodic and trigonometric function that oscillates between 1 and -1 with a period of $$2\pi$$. This means that the sine function repeats its values every $$2\pi$$ units.

Determine end behavior as x approaches ±∞

Since the sine function is periodic and oscillates forever between 1 and -1, it does not settle to any particular value as $$x \to \pm\infty$$. Therefore, there is no specific end behavior such as approaching 0, ±∞, or any other number.

Identify the asymptotes, if any

The sine function doesn't have asymptotes because it doesn't have any values of x for which the function approaches infinity or negative infinity. The sine function is always bounded by 1 and -1.

Sketch the graph

To sketch the graph of $$f(x) = \sin x$$, begin by marking the x-axis with multiples of $$\frac{\pi}{2}$$ (i.e., $$\frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$$, and so on). The sine function oscillates between 1 and -1 with zeros at every multiple of $$\pi$$. The crest and trough occur at every odd multiple of $$\frac{\pi}{2}$$. Draw a smooth curve that oscillates between 1 and -1, making sure it intersects the x-axis at each multiple of $$\pi$$, and then it reaches its highest and lowest points at every odd multiple of $$\frac{\pi}{2}$$. Since this behavior repeats indefinitely, a few periods should be sufficient to represent the overall shape of the sine function.

Key concepts

Transcendental Functions

Transcendental functions are an exciting class of functions that go beyond the algebraic functions you're most accustomed to, like polynomials. These include functions with no simple algebraic solution. They transcend typical algebraic operations, as their name suggests. Some common transcendental functions include exponential functions, logarithms, and trigonometric functions like sine and cosine.

Transcendental functions often appear in real-world applications ranging from physics to engineering and sometimes exhibit intriguing behaviors such as periodicity or asymptotic behavior. In our case, the sine function, a transcendental function, provides an excellent example of periodicity and oscillation. It repeats itself over regular intervals, known as its period, without extending to infinity either positively or negatively.

Sine Function

The sine function, written as \(f(x)=\sin x\), is one of the most basic transcendental functions. Characterized by its wave-like shape, it oscillates repeatedly.

• This function has a range of values from -1 to 1.
• It displays a periodic behavior with a period of \(2\pi\).
• The sine function achieves zero at multiples of \(\pi\), peaks at odd multiples of \(\frac{\pi}{2}\), and troughs at negative odd multiples of \(\frac{\pi}{2}\).

Because of its periodic nature, the sine function is perfect for describing cyclical phenomena like sound waves and alternating currents. In mathematics, it's commonly used in Fourier series to represent various forms of wave functions by combining sines and cosines.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. They are significant in analyzing the behavior of functions, especially those with values that extend to infinity. However, the sine function is unique.

• The sine function doesn't have asymptotes.
• This lack of asymptotes is due to its limited range between -1 and 1.
• For the sine function, there isn't a line it strives to reach but never actually does.

In contrast, many other transcendental functions do have asymptotic behavior. For instance, the exponential function has a horizontal asymptote, and logarithmic functions often have vertical asymptotes. Asymptotes play a crucial role in understanding the limits and behavior of those functions as they extend beyond finite values.

Graph Sketching

Graph sketching is an essential skill in understanding functions visually. When you sketch the graph of the sine function, there are key characteristics to remember.

• Start by marking important points: the zeros at each multiple of \(\pi\) and the peaks and troughs at each odd multiple of \(\frac{\pi}{2}\).
• The sine function should smoothly oscillate, reaching maximum and minimum at every odd multiple of \(\frac{\pi}{2}\).
• Due to the function's periodicity, illustrating more than one period provides a more complete visual representation.

The graph of the sine function is a continuous wave, emphasizing its perpetual oscillation. By sketching it over a few periods, you gain a clearer view not just of its repeating pattern, but also an understanding of its use in modeling real-world periodic processes.