Problem 82 - Calculus: Early Transcendentals

Question

End behavior for transcendental functions. Consider the graph of \(y=\cot ^{-1} x\) (see Section 1.4 ) and determine the following limits using the graph. a. \(\lim _{x \rightarrow \infty} \cot ^{-1} x\) b. \(\lim _{x \rightarrow-\infty} \cot ^{-1} x\)

Step-by-Step Solution

Answer

Question: Determine the limits of the inverse cotangent function, \(y=\cot^{-1}x\), as \(x\) approaches positive infinity and negative infinity. Answer: For the inverse cotangent function \(y=\cot^{-1}x\): a. The limit as \(x\) approaches positive infinity is \(0\). b. The limit as \(x\) approaches negative infinity is \(\pi\).

Step 1: Understand the properties of the \(\cot^{-1}x\) function

The inverse cotangent function, \(y=\cot^{-1}x\) is the inverse function of the cotangent function. The cotangent function has a period of \(\pi\) and has asymptotes for all integral multiples of \(\pi/2\). Therefore, the range of the inverse cotangent function will be from \(0\) to \(\pi\) with an asymptote at \(y=\pi/2\). The graph of \(\cot^{-1}x\) will start from the point \((0,\pi)\) as it's an odd function and will approach the horizontal asymptote \(y=0\), as \(x\) approaches \(\infty\). Similarly, the graph will approach the horizontal asymptote \(y=\pi\), as \(x\) approaches \(-\infty\).

Step 2: Determine the limit as \(x\) approaches \(\infty\)

As \(x\) approaches positive infinity, the graph of \(\cot^{-1}x\) gets closer to the horizontal asymptote \(y=0\). Therefore, the limit as \(x\) approaches positive infinity is: $$\lim_{x \rightarrow \infty} \cot^{-1}x = 0$$

Step 3: Determine the limit as \(x\) approaches \(-\infty\)

As \(x\) approaches negative infinity, the graph of \(\cot^{-1}x\) gets closer to the horizontal asymptote \(y=\pi\). Therefore, the limit as \(x\) approaches negative infinity is: $$\lim_{x \rightarrow -\infty} \cot^{-1}x = \pi$$

Step 4: Provide the final answer

After analyzing the graph of the function \(y=\cot^{-1}x\), the requested limits are found to be: a. \(\lim_{x \rightarrow \infty} \cot^{-1}x = 0\) b. \(\lim_{x \rightarrow -\infty} \cot^{-1}x = \pi\)

Key Concepts

End Behavior

In mathematics, end behavior refers to the behavior of a graph of a function as it approaches positive or negative infinity on the x-axis. Understanding end behavior is crucial when analyzing transcendental functions like the inverse trigonometric functions, such as the inverse cotangent function, denoted as \(\cot^{-1}x\).
For the function \(y = \cot^{-1}x\), the end behavior can be deduced from its graph and properties. As \(x\) moves towards positive infinity (\(x \to \infty\)), the function approaches zero. This is because the graph comes closer to the x-axis, trending towards the horizontal asymptote at \(y = 0\).
Conversely, as \(x\) approaches negative infinity (\(x \to -\infty\)), the graph of \(\cot^{-1}x\) approaches a value of \(\pi\). Hence, the graph trends upwards to meet the horizontal asymptote at \(y = \pi\). Understanding the end behavior helps in visualizing how the graph behaves even without plotting points explicitly.

Limits at Infinity

Limits at infinity help us describe the behavior of a function's values as the input grows without bound. For \(y = \cot^{-1}x\), studying limits at infinity provides insights into its asymptotic behaviors and how the function behaves at both ends of the x-axis.
As \(x\) approaches \(+\infty\), the limit of \(\cot^{-1}x\) is determined by its tendency to approach the x-axis. This results in the following relationship:
\[\lim_{x \to \infty} \cot^{-1}x = 0\]
Similarly, considering the behavior as \(x\) approaches \(-\infty\), the inverse cotangent function approaches the value of \(\pi\), showing that the graph extends upwards to meet this horizontal line:
\[\lim_{x \to -\infty} \cot^{-1}x = \pi\]
Understanding and calculating these limits at infinity allow you to predict and confirm the behavior of the function at these extreme values of \(x\).

Asymptotes

Asymptotes are lines that a graph approaches as the input grows very large or very small, providing critical insight into the graphical behavior of functions like \(\cot^{-1}x\). They do not intersect the graph of the function but indicate directions or bounds the graph adheres to.
For \(\cot^{-1}x\), horizontal asymptotes are particularly essential. The graph approaches the asymptote \(y = 0\) as \(x\) heads towards positive infinity. This means the curve flattens and comes close to the x-axis with increasing positive values of \(x\).
On the negative side, as \(x\) moves to \(-\infty\), the graph of \(\cot^{-1}x\) trends toward the horizontal asymptote \(y = \pi\). These asymptotes help determine the ultimate direction in which the graph tends and give a clear picture of the function's end behavior and limits at infinity.