**Title: Understanding Piecewise Functions and Limits**---**Introduction**In this section, we will explore the concept of piecewise functions and their graphical representation. We will also look into how to evaluate the limit of such functions as they approach a specific point.---**Piecewise Function Example**Consider the following piecewise function:\[f(x) = \begin{cases} x + 5, & \text{if } x \leq 3 \\ -2x - 1, & \text{if } x > 3 \end{cases}\]**Task**We are tasked with sketching the graph of this function and evaluating the limit of $ f(x) $ as $ x $ approaches 3.---**Graphical Representation**- **Part 1**: For $ x \leq 3 $, the function is $ f(x) = x + 5 $. - This is a straight line with a slope of 1 and a y-intercept at 5. - The line continues until $ x = 3 $, where it is represented as a solid dot indicating that the point is included in this segment.- **Part 2**: For $ x > 3 $, the function becomes $ f(x) = -2x - 1 $. - This is a line with a slope of -2 and a y-intercept at -1. - The graph begins at $ x = 3 $, shown as an open circle to indicate that this point is not included.**Graph Detail Explanation**- The graph shows two distinct linear parts for the function.- As $ x $ increases up to 3, the line follows the equation $ y = x + 5 $, reaching a value of 8 at $ x = 3 $.- Beyond $ x = 3 $, the line shifts to follow $ y = -2x - 1 $, beginning at the open circle above $ x = 3 $.---**Evaluating the Limit**To find $\lim_{x \to 3} f(x)$, we need to consider the behavior of the function as $ x $ approaches 3 from both sides:- As $ x $ approaches 3 from the left ($ x \leq 3 $), $ f(x) = x + 5 $ approaches

Given, f(x)=x+5, if x≤-3-2x-1, if x>-3

Answered: Sketch the graph of the function f and…

Math

Calculus

Sketch the graph of the function f and evaluate lim f(x). (x+5, if x s-3 f(x) = %3D -2x-1, ifx >-3

Intermediate Algebra

10th Edition

ISBN:9781285195728

Author:Jerome E. Kaufmann, Karen L. Schwitters

Publisher:Jerome E. Kaufmann, Karen L. Schwitters

Chapter9: Functions

Section9.1: Relations And Functions

Problem 75PS

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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

**Title: Understanding Piecewise Functions and Limits**

---

**Introduction**

In this section, we will explore the concept of piecewise functions and their graphical representation. We will also look into how to evaluate the limit of such functions as they approach a specific point.

---

**Piecewise Function Example**

Consider the following piecewise function:

\[
f(x) =
\begin{cases}
x + 5, & \text{if } x \leq 3 \\
-2x - 1, & \text{if } x > 3
\end{cases}
\]

**Task**

We are tasked with sketching the graph of this function and evaluating the limit of $ f(x) $ as $ x $ approaches 3.

---

**Graphical Representation**

- **Part 1**: For $ x \leq 3 $, the function is $ f(x) = x + 5 $.
- This is a straight line with a slope of 1 and a y-intercept at 5.
- The line continues until $ x = 3 $, where it is represented as a solid dot indicating that the point is included in this segment.

- **Part 2**: For $ x > 3 $, the function becomes $ f(x) = -2x - 1 $.
- This is a line with a slope of -2 and a y-intercept at -1.
- The graph begins at $ x = 3 $, shown as an open circle to indicate that this point is not included.

**Graph Detail Explanation**

- The graph shows two distinct linear parts for the function.
- As $ x $ increases up to 3, the line follows the equation $ y = x + 5 $, reaching a value of 8 at $ x = 3 $.
- Beyond $ x = 3 $, the line shifts to follow $ y = -2x - 1 $, beginning at the open circle above $ x = 3 $.

---

**Evaluating the Limit**

To find $\lim_{x \to 3} f(x)$, we need to consider the behavior of the function as $ x $ approaches 3 from both sides:

- As $ x $ approaches 3 from the left ($ x \leq 3 $), $ f(x) = x + 5 $ approaches

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