/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 2 Give an example of a function th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 2

Give an example of a function that has a right-hand limit but not a left-hand limit at a point.

Short Answer

Expert verified
The function \(f(x) = \frac{1}{x}\) for \(x > 0\) and \(f(x) = 0\) for \(x \leq 0\) is an example of a function having a right-hand limit but no left-hand limit at the point 0.

Step by step solution

01

Concept of Limits

Understand the concept of limits. A right-hand limit of a function at a point is the value that the function approaches as the variables approach that point from the right (or higher values). A left-hand limit is the value the function approaches as the variable approach from the left (or lower values).
02

Formulating an Example

Let's take the following function: \(f(x) = \frac{1}{x}\) for \(x > 0\) and \(f(x) = 0\) for \(x \leq 0\). The function \(f(x)\) approaches \(+\infty\) as \(x\) approaches 0 from the right, so the right-hand limit exists. However, as \(x\) approaches 0 from the left, \(f(x)\) is a constant 0, which does not approach \(+\infty\), hence the left-hand limit does not exist at the point.
03

Verifying the function

Verify the function based on the criteria. The function \(f(x) = \frac{1}{x}\) for \(x > 0\) and \(f(x) = 0\) for \(x \leq 0\) indeed approaches \(+\infty\) as \(x\) approaches 0 from the right, hence it meets the requirement of having a right-hand limit. As \(x\) approaches 0 from the left, \(f(x)\) is a constant 0, which doesn't approach \(+\infty\), so there is no left-hand limit, fulfilling the requirement of not having a left-hand limit.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Right-hand limit
In calculus, understanding right-hand limits is essential when examining how a function behaves as its input approaches a specific point from the right. Technically, the right-hand limit of a function at a point, say \( x = a \), is denoted by \( \lim_{{x \to a^+}} f(x) \). This notation indicates that you are interested in values of \( x \) that are slightly greater than \( a \).

For the function \( f(x) = \frac{1}{x} \) when \( x > 0 \), as \( x \) gets closer to zero from the positive side, \( f(x) \) approaches infinity. This behavior illustrates the existence of a right-hand limit at \( x = 0 \), showing us that the function is increasing rapidly without bound as it nears the point from the right.

When you examine functions for right-hand limits, look for:
  • Values approaching a specific constant or infinity.
  • Behavior from values greater than the point in question.
Right-hand limits often help in understanding discontinuities and defining behavior at endpoints of functions.
Left-hand limit
Left-hand limits are concerned with the behavior of a function as the input approaches a point from the left, or the side of smaller values. Analogous to right-hand limits, the left-hand limit is expressed as \( \lim_{{x \to a^-}} f(x) \). This indicates a focus on values slightly less than \( a \).

In the case of the function \( f(x) = \frac{1}{x} \) for \( x > 0 \) and \( f(x) = 0 \) for \( x \leq 0 \), as \( x \) approaches zero from the left, \( x \leq 0 \), the function remains constant at zero. There is no transition or approach to infinity, hence, no left-hand limit exists at the point \( x = 0 \).

When determining left-hand limits, consider:
  • The value or trend observed when the point is approached from the negative side.
  • Consistency or change of value on this side of the point.
The lack of a left-hand limit highlights abrupt changes or discontinuities in function behavior.
Function behavior
Function behavior refers to the overall trend and characteristics of a function as its inputs vary across its domain. It includes examining how functions approach values at specific points, particularly through limits from both directions.

In the given example of the function \( f(x) = \frac{1}{x} \) for \( x > 0 \), and \( f(x) = 0 \) for \( x \leq 0 \), the behavior showcases a critical instance of differentiating right and left-hand limits. From the right, the function shoots towards infinity indicating that values increase without bound, while from the left, the function remains stagnant at zero.

Studying function behavior is crucial to:
  • Identifying points of discontinuity and undefined behavior.
  • Understanding the continuity and differentiability of the function.
  • Exploring the mathematical and real-world implications of these behaviors.
Thus, analyzing the function behavior via limits provides deeper insights into the characteristics and potential anomalies of the given functions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Prove that \(\lim _{x \rightarrow 0} \cos (1 / x)\) does not exist but that \(\lim _{x \rightarrow 0} x \cos (1 / x)=0\).

Evaluate the following limits, or show that they do not exist. (a) \(\lim _{x \rightarrow 1+} \frac{x}{x-1} \quad(x \neq 1)\), (b) \(\lim _{x \rightarrow 1} \frac{x}{x-1} \quad(x \neq 1)\), (c) \(\lim _{x \rightarrow 0+}(x+2) / \sqrt{x} \quad(x>0)\), (d) \(\lim _{x \rightarrow \infty}(x+2) / \sqrt{x} \quad(x>0)\), (e) \(\lim _{x \rightarrow 0}(\sqrt{x+1}) / x \quad(x>-1)\), (f) \(\lim _{x \rightarrow \infty}(\sqrt{x+1}) / x \quad(x>0)\), (g) \(\lim _{x \rightarrow \infty} \frac{\sqrt{x}-5}{\sqrt{x}+3} \quad(x>0)\), (h) \(\lim _{x \rightarrow \infty} \frac{\sqrt{x}-x}{\sqrt{x}+x} \quad(x>0)\).

Use either the \(\varepsilon-\delta\) definition of limit or the Sequential Criterion for limits, to establish the following limits. (a) \(\lim _{x \rightarrow 2} \frac{1}{1-x}=-1\), (b) \(\lim _{x \rightarrow 1} \frac{x}{1+x}=\frac{1}{2}\), (c) \(\lim _{x \rightarrow 0} \frac{x^{2}}{|x|}=0\), (d) \(\lim _{x \rightarrow 1} \frac{x^{2}-x+1}{x+1}=\frac{1}{2}\).

Show that \(\lim _{x \rightarrow c} x^{3}=c^{3}\) for any \(c \in \mathbb{R}\).

Suppose the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) has limit \(L\) at 0, and let \(a>0\). If \(g: \mathbb{R} \rightarrow \mathbb{R}\) is defined by \(g(x):=f(a x)\) for \(x \in \mathbb{R}\), show that \(\lim _{x \rightarrow 0} g(x)=L .\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.