Question

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

Short answer

Question: Describe the end behavior of the function $$f(x) = \sin(x)$$, and provide a simple sketch of its graph. Answer: The sine function $$f(x) = \sin(x)$$ does not have a limit as x approaches positive or negative infinity, so its end behavior is undefined. A simple sketch of the graph shows that it oscillates between -1 and 1, with a period of $$2\pi$$, and has zeros at all integer multiples of $$\pi$$. The graph has a peak at $$x = \frac{\pi}{2}$$ with a value of $$1$$ and a trough at $$x = -\frac{\pi}{2}$$ with a value of $$-1$$.

Step-by-step solution

Evaluate the limits as x approaches infinity

Since the sine function $$f(x) = \sin(x)$$ is periodic and oscillates between -1 and 1, it does not have limits as x approaches positive or negative infinity. Hence, the end behavior is undefined.

Provide a simple sketch of the graph

To sketch the graph of the sine function $$f(x) = \sin(x)$$, we can start by noting the periodic nature of the sine function, with a period of $$2\pi$$. The graph oscillates between -1 and 1 and has zeros at all integer multiples of $$\pi$$. To draw a basic sketch of the graph, follow these guidelines: 1. Mark the x-axis with values $$0$$, $$\pi$$, $$2\pi$$, $$-\pi$$, $$-2\pi$$, and so on. 2. Mark the y-axis with values $$-1$$ and $$1$$. 3. At each integer multiple of $$\pi$$, the sine function intersects the x-axis. 4. Between $$0$$ and $$\pi$$, the sine function has a peak at $$x = \frac{\pi}{2}$$ with a value of $$1$$. 5. Between $$0$$ and $$-\pi$$, the sine function has a trough at $$x = -\frac{\pi}{2}$$ with a value of $$-1$$. Once you have used the above guidelines to sketch the graph, you should have a clear understanding of the behavior of the sine function. Note that there are no asymptotes since the function oscillates between -1 and 1 and does not approach any fixed values as x approaches infinity.

Key concepts

Sine Function

The sine function, denoted as \( f(x) = \sin(x) \), is a fundamental trigonometric function with widespread applications. Its main property is periodicity, meaning it repeats its pattern at regular intervals. The sine function oscillates smoothly between a maximum value of 1 and a minimum value of -1. This oscillation is continuous, making the sine wave a smooth, undulating line. The period of the sine function is \( 2\pi \), which means that after every \( 2\pi \) units along the x-axis, the function starts a new cycle.

• The sine function is defined for all real numbers, making it a continuous function.
• It is an odd function, satisfying \( \sin(-x) = -\sin(x) \).
• The sine wave crosses the x-axis at integer multiples of \( \pi \), where it takes the value of 0.

These basic properties make the sine function integral to understanding wave behavior, sound, and light, among other phenomena.

Graph Sketching

Sketching the graph of the sine function involves understanding its periodic nature and characteristic points. The graph is a wave that oscillates above and below the x-axis. To begin sketching:

• Mark the x-axis at key points: \( 0, \pi, 2\pi, -\pi, -2\pi \).
• Identify the y-values at these x-points: 0 at \( k\pi \), with peaks at \( (2k+1)\pi/2 \) resulting in \( 1 \) and troughs resulting in \( -1 \) at \( (2k-1)\pi/2 \).

The intermediate peaks and troughs of 1 and -1 between these zeros correspond to \( x=\pi/2 \) and \( x=-\pi/2 \), respectively. By connecting these points with a smooth and continuous curve, you capture the essence of the sine wave. The resulting graph is symmetrical around the origin, reflecting its odd nature.

End Behavior

The concept of end behavior in functions typically describes how a function behaves as its input values become very large or very small. For the sine function \( f(x) = \sin(x) \), the end behavior is unique due to its periodicity:

• As \( x \to \infty \) or \( x \to -\infty \), the sine function continues to oscillate between -1 and 1.
• This means it does not approach a specific value but rather continuously cycles through its range.

Unlike polynomial or rational functions, the sine function's end behavior cannot be captured with a simple limit. Instead, its constant oscillation is the key feature of its behavior at infinity, illustrating why understanding transcendental functions requires a special approach.

Asymptotes

The sine function, \( f(x) = \sin(x) \), does not have asymptotes. Asymptotes occur in graphs where the function approaches a specific line but never meets it, often found in rational functions or logarithmic graphs.
For the sine function:

• There are no horizontal asymptotes because the function does not approach a fixed y-value as \( x \to \pm \infty \).
• There are no vertical asymptotes since \( \sin(x) \) is defined for all real \( x \), without any undefined singularities or asymptotic behavior.

The lack of asymptotes in the sine function reinforces its periodic and bounded nature. This is crucial for sinusoidal functions in modeling oscillatory phenomena, as it allows for predictions of future behavior without divergence to infinity.