Problem 4 In each part, find the stated li... [FREE SOLUTION]

Chapter 2: Problem 4

In each part, find the stated limit of \(f(x)=x /|x|\) by inspection. (a) \(\lim _{x \rightarrow 5} f(x)\) (b) \(\lim _{x \rightarrow-5} f(x)\) (c) \(\lim _{x \rightarrow+\infty} f(x)\) (d) \(\lim f(x)\) (e) \(\lim _{x \rightarrow 0^{+}} f(x)\) (f) \(\lim _{x \rightarrow 0^{-}} f(x)\)

Short Answer

Expert verified

(a) 1, (b) -1, (c) 1, (d) DNE, (e) 1, (f) -1.

Step by step solution

Understand the Function

The function given is \(f(x) = \frac{x}{|x|}\). This function equals 1 if \(x > 0\) and -1 if \(x < 0\). The absolute value in the denominator only affects the sign of \(x\) when determining the function's value.

Evaluate Limit as x Approaches Positive 5

For \(\lim_{x \to 5} f(x)\), since 5 is positive, \(|x| = x\), making \(f(x) = \frac{x}{x} = 1\). The limit is 1.

Evaluate Limit as x Approaches Negative 5

For \(\lim_{x \to -5} f(x)\), because -5 is negative, \(|x| = -x\), making \(f(x) = \frac{x}{-x} = -1\). The limit is -1.

Evaluate Limit as x Approaches Positive Infinity

As \(x\) approaches \(+\infty\), \(x\) remains positive. Therefore, \(f(x) = 1\). The limit is 1.

Evaluate Overall Limit If It Exists

The limit \(\lim f(x)\) does not exist because \(f(x)\) does not approach a single value as \(x\) approaches any direction without specification.

Evaluate Limit as x Approaches Zero from the Positive Side

For \(\lim_{x \to 0^+} f(x)\), since \(x\) approaches zero from the positive side, \(f(x) = 1\). The limit is 1.

Evaluate Limit as x Approaches Zero from the Negative Side

For \(\lim_{x \to 0^-} f(x)\), as \(x\) approaches zero from the negative side, \(f(x) = -1\). The limit is -1.

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Key Concepts

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

Limit Analysis

Limit analysis is a fundamental concept in calculus. It involves evaluating the behavior of a function as the input approaches a particular value. In this exercise, we study several limits of the function \( f(x) = \frac{x}{|x|} \). To analyze limits, we consider both the value the variable \( x \) approaches and the direction from which it approaches this value. By determining how the function responds as \( x \) nears different points, we can gain insights into its overall behavior and continuity. When performing limit analysis, the key steps are:

Identify the point \( x \) approaches
Determine if there are one-sided limits
Assess the function's behavior as \( x \) approaches infinity or negative infinity
Check if a limit exists or does not exist

Piecewise Functions

Piecewise functions are composed of multiple sub-functions, each applying to a part of the main function's domain. They allow different rules for different input values. In this exercise, the function \( f(x) \) can be seen as a piecewise function since:

For \( x > 0 \), \( f(x) = 1 \)
For \( x < 0 \), \( f(x) = -1 \)

The behavior of \( f(x) \) changes based on which inequality \( x \) satisfies. This switch at \( x = 0 \) makes piecewise functions very useful for modeling real-world situations where a condition causes a change in outcome. The function is undefined at \( x = 0 \), highlighting a typical property of piecewise functions: they may not be continuous at transition points.

Infinite Limits

Infinite limits arise when a function's values increase or decrease without bound as \( x \) approaches a particular value. For \( f(x) \), infinite limits are considered when \( x \) tends to very large numbers. In this exercise:

As \( x \rightarrow +\infty \), \( f(x) = 1 \)
As \( x \rightarrow -\infty \), \( f(x) = -1 \)

It's crucial to notice that infinity is not a number but a concept to describe behavior. When you say a function "approaches infinity," you mean its magnitude grows indefinitely. Understanding infinite limits adds to our comprehension of a function's end behavior.

One-Sided Limits

When evaluating limits, sometimes a function behaves differently depending on the direction from which \( x \) approaches a point. These are known as one-sided limits. For the function \( f(x) = \frac{x}{|x|} \), one-sided limits help to understand what happens as \( x \) approaches zero. Specifically:

The right-hand limit, \( \lim_{x \rightarrow 0^+} f(x) = 1 \), indicates behavior as \( x \) nears 0 from the positive side.
The left-hand limit, \( \lim_{x \rightarrow 0^-} f(x) = -1 \), shows behavior as \( x \) comes from the negative side.

These limits are crucial when exploring scenarios where a function may not be continuous at a specific point. If the one-sided limits from both directions aren't the same, it indicates a jump discontinuity at that point.

91影视

In each part, find the stated limit of \(f(x)=x /|x|\) by inspection. (a) \(\lim _{x \rightarrow 5} f(x)\) (b) \(\lim _{x \rightarrow-5} f(x)\) (c) \(\lim _{x \rightarrow+\infty} f(x)\) (d) \(\lim f(x)\) (e) \(\lim _{x \rightarrow 0^{+}} f(x)\) (f) \(\lim _{x \rightarrow 0^{-}} f(x)\)

Short Answer

Step by step solution

Understand the Function

Evaluate Limit as x Approaches Positive 5

Evaluate Limit as x Approaches Negative 5

Evaluate Limit as x Approaches Positive Infinity

Evaluate Overall Limit If It Exists

Evaluate Limit as x Approaches Zero from the Positive Side

Evaluate Limit as x Approaches Zero from the Negative Side

Key Concepts

Limit Analysis

Piecewise Functions

Infinite Limits

One-Sided Limits

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