/*! 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 32 One-Sided Limits Graph the piece... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 32

One-Sided Limits Graph the piecewise-defined function and use your graph to find the values of the limits, if they exist. $$f(x)=\left\\{\begin{array}{ll} 2 x+10 & \text { if } x \leq-2 \\ -x+4 & \text { if } x>-2 \end{array}\right.$$ (a) \(\lim _{x \rightarrow-2^{-}} f(x)\) (b) \(\lim _{x \rightarrow-2^{+}} f(x)\) (c) \(\lim _{x \rightarrow-2} f(x)\)

Short Answer

Expert verified
The limits are (a) 6, (b) 6, (c) 6.

Step by step solution

01

Understanding the Functions

The function is piecewise-defined with two sections: - For \( x \leq -2 \), the function is \( f(x) = 2x + 10 \).- For \( x > -2 \), the function is \( f(x) = -x + 4 \).
02

Graphing the Left Function

Graph \( f(x) = 2x + 10 \) for \( x \leq -2 \). Calculate the endpoint:\[ x = -2: \quad f(-2) = 2(-2) + 10 = 6 \].As \( x \to -\infty \), the graph follows the line \( f(x) = 2x + 10 \). Mark the point \((-2, 6)\) as solid since it's defined in this piece.
03

Graphing the Right Function

Graph \( f(x) = -x + 4 \) for \( x > -2 \). Find a starting point:\[ x = -2: \quad f(-2)(approach from right) = -(-2) + 4 = 6 \].This point \((-2, 6)\) is not part of this graph, so mark it on the graph with an open circle to distinguish it from the defined left piece.
04

Evaluating the Left-Sided Limit

Evaluate \( \lim_{x \to -2^-} f(x) \) using the left segment \( f(x) = 2x + 10 \):Calculate: \[ \lim_{x \to -2^-} (2x + 10) = 2(-2) + 10 = 6 \].Therefore, \( \lim_{x \to -2^-} f(x) = 6 \).
05

Evaluating the Right-Sided Limit

Evaluate \( \lim_{x \to -2^+} f(x) \) using the right segment \( f(x) = -x + 4 \):Calculate: \[ \lim_{x \to -2^+} (-x + 4) = 2 + 4 = 6 \].Thus, \( \lim_{x \to -2^+} f(x) = 6 \).
06

Evaluating the Two-Sided Limit

Check if both one-sided limits are equal. Since \( \lim_{x \to -2^-} f(x) = 6 \) and \( \lim_{x \to -2^+} f(x) = 6 \),\[ \lim_{x \to -2} f(x) = 6 \]. The two-sided limit exists and is 6.

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 composed of different expressions based on different parts of the domain. It means that the function behaves differently depending on the interval of the input values. For example, in the given exercise, the function is defined by two expressions:
  • For inputs where \( x \leq -2 \), the function is \( f(x) = 2x + 10 \).
  • For inputs where \( x > -2 \), the function is \( f(x) = -x + 4 \).
The key idea is that you are tentatively switching between these expressions based on the value of \( x \). When dealing with piecewise functions, it's critical to pay attention to the inequalities that define each section. This is what guides you in determining how the function behaves over various parts of its domain.
Graphing Functions
Graphing a piecewise-defined function involves plotting each segment separately on a coordinate plane and paying attention to the domains they cover. When graphing:
  • Determine the equations used for each piece and the intervals over which they apply.
  • Plot only the relevant segment of the graph according to its interval, using solid or open dots to indicate whether endpoints are included.
For the given example:- For \( x \leq -2 \), graph \( f(x) = 2x + 10 \) ensuring a solid dot at \((-2, 6)\) as it includes \( x = -2 \).
- For \( x > -2 \), graph \( f(x) = -x + 4 \) starting just after \( x = -2 \) and using an open dot to show that the function does not include \( x = -2 \).
Graphing accurately is crucial as it visually illustrates how functions behave across different intervals of the domain.
Evaluating Limits
Evaluating limits involves determining what value a function approaches as the input gets infinitely close to a particular point. Sometimes, this involves checking from both sides (left and right) to get a complete picture. Here's how it's done:
  • For the left-sided limit \( \lim_{x \to -2^-} f(x) \), you consider only the values approaching \( -2 \) from the left (i.e., \( x \leq -2 \)).
  • For the right-sided limit \( \lim_{x \to -2^+} f(x) \), use the function values moving towards \( -2 \) from the right (i.e., \( x > -2 \)).
In both cases, it's essential to substitute values close to \( x = -2 \) into the respective function expressions. If the results from both sides match, it indicates that the overall limit at that point is the same and exists, confirming continuity at that point.
Two-Sided Limit
A two-sided limit at a point occurs when the left-sided limit and the right-sided limit at that point are equal. In the exercise, this is verified by comparing the results from evaluating limits at \( x = -2 \) from both sides:
  • The left-sided limit, \( \lim_{x \to -2^-} f(x) \), is \( 6 \).
  • The right-sided limit, \( \lim_{x \to -2^+} f(x) \), is \( 6 \) too.
Since both are equal, \( \lim_{x \to -2} f(x) \) is \( 6 \), and hence, the two-sided limit exists. Identifying two-sided limits helps confirm function continuity at specific points and ensures that the function doesn't have jumps or breaks at those points.

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

Tangent Lines (a) If \(g(x)=1 /(2 x-1),\) find \(g^{\prime}(a)\) (b) Find equations of the tangent lines to the graph of \(g\) at the points whose \(x\) -coordinates are \(-1,0,\) and 1 (c) Graph \(g\) and the three tangent lines.

Find the slope of the tangent line to the graph of \(f\) at the given point. $$f(x)=\frac{6}{x+1}, \quad \text { at }(2,2)$$

Find the derivative of the function at the given number. $$f(x)=x-3 x^{2}, \quad \text { at }-1$$

Find the limit, and use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 1} \frac{x^{2}-1}{\sqrt{x}-1}$$

Estimating Limits Numerically and Graphically Estimate the value of the limit by making a table of values. Check your work with a graph. $$\lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x-3}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Decision Maths

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.