Certainly! Below is the transcription and explanation for an educational website:---### Problem 2#### Let $ f(x) = \begin{cases} x^2 & \text{if } x \neq 2 \\ 5 & \text{if } x = 2 \end{cases} $**a. Find $ \lim_{{x \to 2}} f(x) $.****b. Find $ f(2) $.****c. Is $ f(x) $ continuous at $ x = 2 $? Explain.****d. Sketch a graph of $ f(x) $.**---#### Explanation:This problem involves a piecewise function $ f(x) $ which has different definitions depending on the value of $ x $. Specifically:- For all $ x $ values not equal to 2, the function is defined as $ f(x) = x^2 $.- For $ x $ equal to 2, the function takes on the value $ f(2) = 5 $.#### Steps to Solve:**a. Finding $ \lim_{{x \to 2}} f(x) $:**To determine the limit of $ f(x) $ as $ x $ approaches 2, we consider the value of the function as $ x $ gets arbitrarily close to 2, but not equal to 2. Since the function $ f(x) = x^2 $ for all $ x \neq 2 $:\[ \lim_{{x \to 2}} f(x) = \lim_{{x \to 2}} x^2 = 2^2 = 4 \]**b. Finding $ f(2) $:**From the definition of the piecewise function:\[ f(2) = 5 \]**c. Is $ f(x) $ continuous at $ x = 2 $? Explain:**To determine if $ f(x) $ is continuous at $ x = 2 $, we check the following:1. The limit of $ f(x) $ as $ x $ approaches 2 exists: $ \lim_{{x \to 2}} f(x) = 4 $2. $ f(2) $ is defined: $ f(2) = 5 $3.

Answered: Sx² 15 a. Find lim f(x). if x + 2 2.…

Sx² 15 a. Find lim f(x). if x + 2 2. Let f(x) = } if x = 2 X→2 b. Find f(2). Is f(x) continuous at x = 2? с. Explain. d. Sketch a graph of f(x).

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Concept explainers

Rate of Change

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Certainly! Below is the transcription and explanation for an educational website:

---

### Problem 2

#### Let $ f(x) = \begin{cases}
x^2 & \text{if } x \neq 2 \\
5 & \text{if } x = 2
\end{cases} $

**a. Find $ \lim_{{x \to 2}} f(x) $.**

**b. Find $ f(2) $.**

**c. Is $ f(x) $ continuous at $ x = 2 $? Explain.**

**d. Sketch a graph of $ f(x) $.**

---

#### Explanation:

This problem involves a piecewise function $ f(x) $ which has different definitions depending on the value of $ x $. Specifically:

- For all $ x $ values not equal to 2, the function is defined as $ f(x) = x^2 $.
- For $ x $ equal to 2, the function takes on the value $ f(2) = 5 $.

#### Steps to Solve:

**a. Finding $ \lim_{{x \to 2}} f(x) $:**

To determine the limit of $ f(x) $ as $ x $ approaches 2, we consider the value of the function as $ x $ gets arbitrarily close to 2, but not equal to 2.

Since the function $ f(x) = x^2 $ for all $ x \neq 2 $:
\[ \lim_{{x \to 2}} f(x) = \lim_{{x \to 2}} x^2 = 2^2 = 4 \]

**b. Finding $ f(2) $:**

From the definition of the piecewise function:
\[ f(2) = 5 \]

**c. Is $ f(x) $ continuous at $ x = 2 $? Explain:**

To determine if $ f(x) $ is continuous at $ x = 2 $, we check the following:
1. The limit of $ f(x) $ as $ x $ approaches 2 exists: $ \lim_{{x \to 2}} f(x) = 4 $
2. $ f(2) $ is defined: $ f(2) = 5 $
3.

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