How do I evaluate the limits of 1c, 1d, 1f? **Transcription for Educational Website**---## Understanding Piecewise Functions and Limits: An Analysis of $y = f(x)$### Given Function and Key Points:The provided graph represents the function $ y = f(x) $. To understand the behavior of the function, focus on specific points and limits as marked by black and white dots on the graph.### Graph Analysis:1. **Axes and Grid:** - The $ x $-axis ranges from $-1$ to 5. - The $ y $-axis ranges from 1 to 5. - Major grid lines at integer intervals provide a clear view of function values at those points.2. **Key Points on the Graph:** - Open circles (○) represent points where the value is not included in the function at that specific $ x $-value. - Solid circles (•) represent points where the value is included in the function at that specific $ x $-value.### Specific Queries:Here are the specific questions regarding the function $ y = f(x) $:1. **(a) $ f(1) $:** - From the graph, at $ x = 1 $, there is an open circle at $ y = 2 $. This indicates $ f(1) $ does not exist at $ y = 2 $. Instead, look for a solid circle to define $ f(1) $. Hence, $ f(1) = 3 $.2. **(b) $ \lim_{x \to 1} f(x) $:** - The limit as $ x $ approaches 1 from both sides should meet at a specific y-value. From the function behavior, as $ x \to 1 $, $ f(x) \to 2 $. Therefore, $ \lim_{x \to 1} f(x) = 2 $.3. **(c) $ f(2) $:** - Directly from the graph, $ f(2) = 4 $.4. **(d) $ \lim_{x \to 2} $:** - Observe the function’s approach as $ x $ nears 2. The function values approach the y-value at $ x = 2 $, hence \( \lim_{x \to 2} f(x) = 4 \

Answered: Image /qna-images/answer/456099f2-cd0d-4427-8992-84eae1158785.jpg

Answered: (a)f(1), (6) lim→1 f(x), (c)f (2), (d)…

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(a)f(1), (6) lim→1 f(x), (c)f (2), (d) lim,→2, (e)f(3), (f) lim,→3 yA y = f(x) 4 3. -1 1 2 3 4

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

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

How do I evaluate the limits of 1c, 1d, 1f?

**Transcription for Educational Website**

---

## Understanding Piecewise Functions and Limits: An Analysis of $y = f(x)$

### Given Function and Key Points:

The provided graph represents the function $ y = f(x) $. To understand the behavior of the function, focus on specific points and limits as marked by black and white dots on the graph.

### Graph Analysis:

1. **Axes and Grid:**
- The $ x $-axis ranges from $-1$ to 5.
- The $ y $-axis ranges from 1 to 5.
- Major grid lines at integer intervals provide a clear view of function values at those points.

2. **Key Points on the Graph:**
- Open circles (○) represent points where the value is not included in the function at that specific $ x $-value.
- Solid circles (•) represent points where the value is included in the function at that specific $ x $-value.

### Specific Queries:

Here are the specific questions regarding the function $ y = f(x) $:

1. **(a) $ f(1) $:**
- From the graph, at $ x = 1 $, there is an open circle at $ y = 2 $. This indicates $ f(1) $ does not exist at $ y = 2 $. Instead, look for a solid circle to define $ f(1) $. Hence, $ f(1) = 3 $.

2. **(b) $ \lim_{x \to 1} f(x) $:**
- The limit as $ x $ approaches 1 from both sides should meet at a specific y-value. From the function behavior, as $ x \to 1 $, $ f(x) \to 2 $. Therefore, $ \lim_{x \to 1} f(x) = 2 $.

3. **(c) $ f(2) $:**
- Directly from the graph, $ f(2) = 4 $.

4. **(d) $ \lim_{x \to 2} $:**
- Observe the function’s approach as $ x $ nears 2. The function values approach the y-value at $ x = 2 $, hence \( \lim_{x \to 2} f(x) = 4 \

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