Problem 18 - Calculus of a Single Variable

Question

In Exercises 17–22, find the limit. $$ \lim _{x \rightarrow-\infty} \tanh x $$

Step-by-Step Solution

Answer

\(\lim _{x \rightarrow-\infty} \tanh x = -1\)

Step 1: Understand the Hyperbolic Tangent Function

The hyperbolic tangent function, notated as \(\tanh(x)\), can exhibit two possible horizontal asymptotes (dependent on whether \(x\) approaches positive or negative infinity). As \(x\) approaches positive infinity, \(\tanh(x)\) approaches 1 and as \(x\) approaches negative infinity, \(\tanh(x)\) approaches -1. Importantly, this function is always bounded between -1 and 1.

Step 2: Applying Limit to \(\tanh(x)\)

We now apply the limit as \(x\) approaches negative infinity. Given what we understand about the behavior of the function as \(x\) approaches negative infinity, the function is approaching -1. Therefore, \(\lim _{x \rightarrow-\infty} \tanh x = -1\).

Key Concepts

Limits at Infinity

In calculus, limits at infinity help us understand how functions behave as variable values grow larger or smaller indefinitely. When you evaluate a limit as the variable approaches infinity or negative infinity, you focus on the end behavior of the function.

The concept is crucial for identifying the long-term trends of functions and understanding whether they settle at a specific value or diverge. With the hyperbolic tangent function, \ \( \tanh(x) \ \), as an example, evaluating its limit at infinity or negative infinity reveals whether there's a horizontal line that the function gets arbitrarily close to without ever touching.

When examining \ \( \lim_{x \to -\infty} \tanh(x) \ \), the function approaches -1, indicating that, as x declines towards negative infinity, \( \tanh(x) \) gets closer and closer to -1.

Key takeaways for evaluating limits at infinity include:

• Identifying any horizontal asymptotes.
• Understanding the end behavior of the function.
• Determining if the function converges to a specific value.

Asymptotic Behavior

Asymptotic behavior in mathematics describes how a function behaves as the input approaches a certain value, often infinity. Understanding this behavior is key to identifying limits and horizontal asymptotes.

A horizontal asymptote is a horizontal line that the graph of a function approaches as \(x\) tends towards infinity in either direction. For the function \( \tanh(x) \), horizontal asymptotes exist at \(y = 1\) and \(y = -1\).

• As \(x\) approaches positive infinity, \( \tanh(x) \) approaches 1.
• As \(x\) approaches negative infinity, \( \tanh(x) \) approaches -1.

This asymptotic behavior implies that the values of the function will never exceed \(1\) or fall below \(-1\), regardless of how large \(x\) becomes in magnitude. These limits help define the shape and evaluation of hyperbolic functions and are crucial in solving calculus problems that involve limits.

Calculus Exercises

Calculus exercises often require you to apply learned concepts like limits and asymptotic behavior to specific functions. This exercise focuses on finding \( \lim_{x \to -\infty} \tanh(x) \), which necessitates both an understanding of limits and the behavior of hyperbolic functions.

Here's how you can approach similar calculus exercises with confidence:

• Break down the function to understand its basic properties.
• Evaluate what happens to the function as \(x\) approaches significant points like infinity or zero.
• Apply the concept of asymptotes to predict end behavior.

Practicing these steps with different functions allows you to gain a deeper, intuitive understanding. With hyperbolic functions, using their characteristic bounded range simplifies predicting limits as they tend towards infinity.