Problem 34 - Calculus Early Transcendentals

Question

15-36 Find the limit. $$ \lim _{x \rightarrow \infty} \tan ^{-1}\left(x^{2}-x^{4}\right) $$

Step-by-Step Solution

Answer

The limit is \( \frac{-\pi}{2} \).

Step 1: Understand the behavior of the expression

We need to find the limit of \( \tan^{-1}(x^2 - x^4) \) as \( x \to \infty \). Observe the expression inside the inverse tangent function, \( x^2 - x^4 \). As \( x \to \infty \), since the highest power term is \( x^4 \), it will dominate the behavior of this expression, making \( x^2 - x^4 \to -\infty \).

Step 2: Recall properties of inverse tangent function

The inverse tangent function, \( \tan^{-1}(x) \), approaches \( \frac{-\pi}{2} \) as \( x \to -\infty \). This is a fundamental property of the behavior of the arctangent function as the input tends toward negative infinity.

Step 3: Apply properties to find the limit

Given that \( x^2 - x^4 \to -\infty \) as \( x \to \infty \) and knowing \( \tan^{-1}(x) \to \frac{-\pi}{2} \) as \( x \to -\infty \), we conclude that \( \lim_{x \to \infty} \tan^{-1}(x^2 - x^4) = \frac{-\pi}{2} \).

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. One of its main tools is the concept of a limit, which allows us to understand the behavior of functions as they approach certain points or infinity. Calculus employs differential and integral calculus, focusing on derivatives and integrals to solve problems.Understanding limits is crucial, as they help determine the behavior of functions without actually reaching certain input values. In this problem, we're finding the limit of an expression as it goes to infinity, a common task in calculus. Limits can describe how functions behave over large domains or near points where they aren’t easily defined.In tackling calculus problems, students often start by examining the expression closely, as we did with the term \( x^2 - x^4 \) in this exercise. Such an analysis helps to determine the dominating components of the expression, which influences the outcome of the limit.

Inverse Trigonometric Functions

Inverse trigonometric functions undo the operations of the basic trigonometric functions like sine, cosine, and tangent. They are crucial in various branches of mathematics, including calculus, for solving equations involving angles.The inverse tangent function, written as \( \tan^{-1}(x) \) or \( \text{arctan}(x) \), is particularly important when dealing with limits and asymptotic behaviors. This function maps real numbers to angles between \( \frac{-\pi}{2} \) and \( \frac{\pi}{2} \).

• When \( x \to \infty \), \( \tan^{-1}(x) \to \frac{\pi}{2} \)
• When \( x \to -\infty \), \( \tan^{-1}(x) \to \frac{-\pi}{2} \)

These properties are instrumental for evaluating limits involving inverse trigonometric functions, especially when the inner expression of the function tends toward positive or negative infinity. For instance, in our exercise, \( \tan^{-1}(x^2 - x^4) \) approaches \( \frac{-\pi}{2} \) as \( x \to \infty \).

Asymptotic Behavior

Asymptotic behavior describes how functions behave as inputs become large in magnitude. Particularly, it concerns what values a function approaches but never quite reaches, either as inputs grow towards infinity or shrink towards zero.When analyzing functions like \( x^2 - x^4 \), identifying the highest power term helps. In asymptotic analysis, the term with the largest exponent dominates as \( x \to \infty \). Here, the term \(x^4\) governs the behavior, making \(x^2 - x^4\) trend towards \(-\infty\).Understanding this helps when evaluating limits, as it allows us to use properties of the inverse function involved, such as \( \tan^{-1}(x) \), to predict the asymptotic behavior. In our problem, recognizing that the expression inside the inverse tangent goes to \(-\infty\) helps determine that the limit is \( \frac{-\pi}{2} \).

Limits at Infinity

Limits at infinity concern the value that a function approaches as the variable within it goes to positive or negative infinity. These limits help understand long-term behavior of functions, important in fields ranging from physics to economics.To calculate a limit at infinity, especially involving complex expressions like \(\tan^{-1}(x^2-x^4)\), one identifies the dominant term. In this case, since \( x^4 \) dominates, \( x^2 - x^4 \to -\infty \).When the expression inside a function like \( \tan^{-1} \) approaches a limit, as \( x \to -\infty \), the inverse tangent approaches its horizontal asymptote at \( \frac{-\pi}{2} \). This understanding allows us to conclude that the limit as \( x \to \infty \) of this expression is indeed \( \frac{-\pi}{2} \). Thus, limit analysis at infinity provides a powerful tool for understanding the global behavior of functions.