### Limit Evaluation Problem#### Problem Statement:Find the limit (if it exists). (If an answer does not exist, enter DNE.)\[ \lim_{{x \to 3}} f(x), \text{ where } f(x) = \begin{cases} x^2 - 4x + 18, & \text{for } x < 3 \\ -x^2 + 4x - 14, & \text{for } x \geq 3 \end{cases} \]#### Instructions:- Evaluate the limit as $x$ approaches 3 for the given piecewise function.- If the limit does not exist, write DNE.#### Graph or Diagram Explanation:No graph or diagram is provided with the problem. The function is piecewise, meaning it is defined by different expressions based on the value of $x$:- For $x < 3$, $f(x) = x^2 - 4x + 18$- For $x \geq 3$, $f(x) = -x^2 + 4x - 14$#### Solution Process:To find $\lim_{{x \to 3}} f(x)$, we need to consider the left-hand limit and the right-hand limit:1. **Left-Hand Limit (as $ x $ approaches 3 from the left, $ x < 3 $):** \[ \lim_{{x \to 3^-}} (x^2 - 4x + 18) \] Plug $ x = 3 $ into the function for $ x < 3 $: \[ 3^2 - 4(3) + 18 = 9 - 12 + 18 = 15 \]2. **Right-Hand Limit (as $ x $ approaches 3 from the right, $ x \geq 3 $):** \[ \lim_{{x \to 3^+}} (-x^2 + 4x - 14) \] Plug $ x = 3 $ into the function for $ x \geq 3 $: \[ -(3^2) + 4(3) - 14 = -9 + 12 - 14 = -11 \]Since the left-hand limit \( (

Detailed explanation mentioned below

Answered: Find the limit (if it exists). (If an…

Find the limit (if it exists). (If an answer does not exist, enter DNE.) x2 - 4x + 18, l-x2 + 4x - 14, X< 3 lim f(x), where f(x) x+3 X2 3

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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

### Limit Evaluation Problem

#### Problem Statement:
Find the limit (if it exists). (If an answer does not exist, enter DNE.)

\[ \lim_{{x \to 3}} f(x), \text{ where } f(x) = \begin{cases}
x^2 - 4x + 18, & \text{for } x < 3 \\
-x^2 + 4x - 14, & \text{for } x \geq 3
\end{cases} \]

#### Instructions:
- Evaluate the limit as $x$ approaches 3 for the given piecewise function.
- If the limit does not exist, write DNE.

#### Graph or Diagram Explanation:
No graph or diagram is provided with the problem. The function is piecewise, meaning it is defined by different expressions based on the value of $x$:

- For $x < 3$, $f(x) = x^2 - 4x + 18$
- For $x \geq 3$, $f(x) = -x^2 + 4x - 14$

#### Solution Process:
To find $\lim_{{x \to 3}} f(x)$, we need to consider the left-hand limit and the right-hand limit:

1. **Left-Hand Limit (as $ x $ approaches 3 from the left, $ x < 3 $):**
\[ \lim_{{x \to 3^-}} (x^2 - 4x + 18) \]
Plug $ x = 3 $ into the function for $ x < 3 $:
\[ 3^2 - 4(3) + 18 = 9 - 12 + 18 = 15 \]

2. **Right-Hand Limit (as $ x $ approaches 3 from the right, $ x \geq 3 $):**
\[ \lim_{{x \to 3^+}} (-x^2 + 4x - 14) \]
Plug $ x = 3 $ into the function for $ x \geq 3 $:
\[ -(3^2) + 4(3) - 14 = -9 + 12 - 14 = -11 \]

Since the left-hand limit \( (

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