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Chapter 3: Problem 32

Determine whether the statement is true or false. Explain your answer. If \(\lim _{x \rightarrow+\infty} f^{\prime}(x)=0\), then the graph of \(y=f(x)\) has a horizontal asymptote.

Short Answer

Expert verified
False: Zero derivative at infinity doesn't imply a horizontal asymptote.

Step by step solution

01

Understand the Concept of Derivative

The derivative, denoted as \(f'(x)\), measures the rate of change of a function \(f(x)\). When \(f'(x) = 0\) as \(x\to \infty\), it means that the slope of the tangent to the graph of \(f(x)\) flattens out to become horizontal eventually as \(x\) gets very large.
02

Review the Definition of Horizontal Asymptote

A horizontal asymptote occurs for the function \(f(x)\), when \(\lim_{x \to \infty} f(x) = L\), where \(L\) is a constant. This means that as \(x\) approaches infinity, the value of \(f(x)\) approaches the constant \(L\).
03

Analyze the Implication of the Given Condition

Given \(\lim_{x \to \infty} f'(x) = 0\), this condition only implies that the rate of change (slope) of \(f(x)\) is approaching zero. However, this does not necessarily mean \(f(x)\) itself approaches a particular constant \(L\). The function could still diverge to infinity or oscillate without settling down.
04

Consider Counterexamples

Consider functions such as \(f(x) = \ln(x)\) or \(f(x) = \sqrt{x}\). For both functions, \(\lim_{x \to \infty} f'(x) = 0\) because their derivatives approach zero. However, neither function approaches a constant value \(L\) as \(x\to \infty\), hence they do not have horizontal asymptotes.
05

Conclude Based on Analysis

Due to the fact that a zero derivative at infinity does not ensure the function approaches a finite limit, the assertion \(\lim_{x \to \infty} f'(x) = 0\) does not imply that \(f(x)\) has a horizontal asymptote. Therefore, the statement is false.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

The Derivative Explained
The derivative is a mathematical tool that helps us understand how a function changes. It tells us the slope or steepness of the function's graph at any given point. In simple terms, if you imagine walking along a graph, the derivative tells you how steep or flat the path is at each step.

For example, if you're climbing a hill and the path gets less steep, the derivative decreases. If the derivative approaches zero as you walk, it means the path is leveling out. When we say \( \lim_{x \to \infty} f'(x) = 0 \), it means that as you continue down the path forever, the path gets flatter and flatter until the slope is zero.
  • "Flat" doesn't mean horizontal鈥攊t just means less steep.
  • The slope or derivative tells us how quickly the function's value changes as we move along the x-axis.
  • If \( f'(x) = 0 \) at some point, the graph has a horizontal tangent at that point, but it doesn't dictate the overall shape of the function.
Understanding Limits
In calculus, a limit describes how a function behaves as it gets close to a particular point or as its input grows very large. Limits help us predict the value that the function approaches, even if it doesn't actually reach that value.

For instance, a common limit expression is \( \lim_{x \to \infty} f(x) \), which looks at what happens to \( f(x) \) as \( x \) becomes infinitely large. If \( f(x) \) approaches a constant number \( L \), that's a sign of a horizontal asymptote.
  • Limits sum up a function's behavior as it approaches infinity or other points.
  • An important aspect of limits is determining if they converge to a specific point or diverge.
  • The statement \( \lim_{x \to \infty} f'(x) = 0 \) doesn't necessarily control \( \lim_{x \to \infty} f(x) \).
Exploring Asymptotic Behavior
Asymptotic behavior describes how a function behaves as its input approaches a specific value, usually infinity. If a function stabilizes or approaches a specific line as it continues, this often indicates an asymptote鈥攁 line the graph gets closer to but never touches.

Horizontal asymptotes are especially interesting. They occur when \( \lim_{x \to \infty} f(x) = L \), meaning the function approaches a constant value \( L \) as \( x \) gets very large.
  • Just because a derivative goes to zero doesn't mean the function has a horizontal asymptote.
  • There are functions where \( f'(x) \to 0 \), like natural log \( f(x) = \ln(x) \), where \( \lim_{x \to \infty} f(x) \) is not a constant.
  • Studying asymptotic behavior helps us predict function behavior without visualizing the entire graph.

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Most popular questions from this chapter

Find the limits. $$ \lim _{x \rightarrow+\infty}[\cos (2 / x)]^{x^{2}} $$

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