Problem 30 One-Sided Limits Graph the piece... [FREE SOLUTION]

91影视

Precalculus: Mathematics for Calculus

James Stewart, Lothar Redlin, Saleem Watson

$Math Studyset 91影视 Explanations$ Math

7 Edition

Chapter 13: Problem 30

One-Sided Limits Graph the piecewise-defined function and use your graph to find the values of the limits, if they exist. $$f(x)=\left\\{\begin{array}{ll} 2 & \text { if } x<0 \\ x+1 & \text { if } x \geq 0 \end{array}\right.$$ (a) $\lim _{x \rightarrow 0^{-}} f(x)$ (b) $\lim _{x \rightarrow 0^{+}} f(x)$ (c) $\lim _{x \rightarrow 0} f(x)$

Short Answer

Expert verified

(a) 2, (b) 1, (c) Does not exist.

Step by step solution

Understanding the Piecewise Function

The piecewise function is defined as follows: $f(x) = 2$ for $x < 0$ and $f(x) = x + 1$ for $x \geq 0$. This means there are two parts we need to understand and analyze for their behaviors near $x = 0$.

Graphing the Function

On a graph, plot two different segments: a horizontal line at $y = 2$ for all values of $x < 0$, and a line with a slope of 1 starting at $y = 1$ when $x = 0$ (since $y = x + 1$ implies $1$ when $x = 0$). Mark the point at $x = 0$ as closed for $x + 1$ (since $0 + 1 = 1$) and open for $y = 2$.

Finding $\lim _{x \rightarrow 0^{-}} f(x)$

Evaluate the left-hand limit where $x$ approaches 0 from the negative side. From the graph, for $x < 0$, $f(x) = 2$. Thus, $\lim _{x \rightarrow 0^{-}} f(x) = 2$.

Finding $\lim _{x \rightarrow 0^{+}} f(x)$

Evaluate the right-hand limit where $x$ approaches 0 from the positive side. From the graph, for $x \geq 0$, $f(x) = x + 1$ which gives $f(0) = 0 + 1 = 1$. So, $\lim _{x \rightarrow 0^{+}} f(x) = 1$.

Determining $\lim _{x \rightarrow 0} f(x)$

Compare the left-hand and right-hand limits. Since $\lim _{x \rightarrow 0^{-}} f(x) = 2$ and $\lim _{x \rightarrow 0^{+}} f(x) = 1$, the two limits are not equal. Therefore, the limit $\lim _{x \rightarrow 0} f(x)$ does not exist.

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.

Piecewise-Defined Function

A piecewise-defined function is a type of function in which different formulas are used to define different parts of its domain. These functions are divided into "pieces," each corresponding to a specific interval of the domain.
For example, consider the function given in the exercise:

For all values of $x < 0$, the function $f(x)$ is defined as a constant, $f(x) = 2$. This part of the function is a horizontal line in the graph.
For all values of $x \geq 0$, the function follows another rule, $f(x) = x + 1$. In this case, the function is linear, with a slope of 1 and a y-intercept of 1.

To fully understand a piecewise-defined function, it's crucial to analyze each piece and consider how they fit together, especially at the boundary of their definitions, such as $x = 0$ in this case.

Graphical Analysis

Graphical analysis involves plotting the function to visually assess its behavior and properties. This is a powerful tool for understanding piecewise-defined functions and their limits.
For the exercise at hand:

First plot a flat, horizontal line at $y = 2$ for all $x < 0$. Since the function is constant in this section, the graph does not change until $x = 0$.
Then, for $x \geq 0$, plot a line with a slope of 1 starting from $y = 1$, based on the equation $f(x) = x + 1$.

At the point $x = 0$, you should indicate that the point for the equation $x + 1$ is closed (filled) and for $y = 2$ it's open (unfilled). This graph helps you identify where the limits might differ as $x$ approaches 0.

Left-Hand Limit

The left-hand limit, denoted $\lim_{x \to c^-} f(x)$, evaluates the behavior of $f(x)$ as $x$ approaches a point $c$ from the left-hand side or the negative direction.
For the problem at hand, we look at how $f(x)$ behaves as $x$ gets closer to 0 from values less than 0:

Based on the graph, we see that for $x < 0$, $f(x) = 2$. Therefore, as $x$ approaches 0 from the left, the function remains constant.
This results in the left-hand limit being $\lim_{x \rightarrow 0^{-}} f(x) = 2$.

The left-hand limit gives us an important insight into the potential discontinuity at the point, which occurs when the left and right limits don't match.

Right-Hand Limit

Conversely, the right-hand limit, expressed as $\lim_{x \to c^+} f(x)$, examines the value that $f(x)$ is approaching as $x$ comes in from the right of $c$.
In this case, we observe the behavior of $f(x)$ as $x$ nears 0 from values greater than or equal to 0:

From our function definition, when $x \geq 0$, $f(x) = x + 1$. Thus, exactly at $x = 0$, the function equals 1.
As a result, the right-hand limit is $\lim_{x \rightarrow 0^{+}} f(x) = 1$.

Comparing the left and right limits helps determine if $f(x)$ has an overall limit at $c$. In this exercise, differing values indicate a discontinuity, implying $\lim_{x \to 0} f(x)$ does not exist.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

Understanding the Piecewise Function

Graphing the Function

Finding \(\lim _{x \rightarrow 0^{-}} f(x)\)

Finding \(\lim _{x \rightarrow 0^{+}} f(x)\)

Determining \(\lim _{x \rightarrow 0} f(x)\)

Key Concepts

Piecewise-Defined Function

Graphical Analysis

Left-Hand Limit

Right-Hand Limit

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Statistics

Geometry

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.