lim g(= x-3

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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### Understanding Limits from a Graph: Function g

The graph of the function \( g \) is shown below. Use the graph to determine the following limits:

**Questions:**
a.) \[ \lim_{x \to 3} g(x) \]

b.) \[ \lim_{x \to 1} g(x) \]

c.) \[ \lim_{x \to \infty} g(x) \]

**Graph Explanation:**

The graph provided shows the function \( g(x) \) plotted on a coordinate plane. The \( x \)-axis ranges from \(-5\) to \(8\) and the \( y \)-axis ranges from \(-8\) to \(8\). A vertical dashed line is drawn at \( x = 3 \).

The graph starts from the left, increasing as it approaches the vertical dashed line from both sides (i.e., as \( x \) approaches \( 3 \) from both the left side and right side). The graph shows that as \( x \) approaches 3, the function \( g(x) \) seems to increase without bound.

**Analysis:**

- **For \( \lim_{x \to 3} g(x) \)**: 
  Observing the graph, as \( x \) approaches \( 3 \), \( g(x) \) increases rapidly towards infinity. Therefore,
  \[ \lim_{x \to 3} g(x) \to \infty \]

- **For \( \lim_{x \to 1} g(x) \)**:
  Observing the graph, as \( x \) approaches \( 1 \), \( g(x) \) appears to be around \( 6 \). Therefore,
  \[ \lim_{x \to 1} g(x) \approx 6 \]

- **For \( \lim_{x \to \infty} g(x) \)**:
  Observing the graph, as \( x \) continues to increase towards positive infinity, \( g(x) \) decreases towards \( 0 \). Therefore,
  \[ \lim_{x \to \infty} g(x) = 0 \]
Transcribed Image Text:### Understanding Limits from a Graph: Function g The graph of the function \( g \) is shown below. Use the graph to determine the following limits: **Questions:** a.) \[ \lim_{x \to 3} g(x) \] b.) \[ \lim_{x \to 1} g(x) \] c.) \[ \lim_{x \to \infty} g(x) \] **Graph Explanation:** The graph provided shows the function \( g(x) \) plotted on a coordinate plane. The \( x \)-axis ranges from \(-5\) to \(8\) and the \( y \)-axis ranges from \(-8\) to \(8\). A vertical dashed line is drawn at \( x = 3 \). The graph starts from the left, increasing as it approaches the vertical dashed line from both sides (i.e., as \( x \) approaches \( 3 \) from both the left side and right side). The graph shows that as \( x \) approaches 3, the function \( g(x) \) seems to increase without bound. **Analysis:** - **For \( \lim_{x \to 3} g(x) \)**: Observing the graph, as \( x \) approaches \( 3 \), \( g(x) \) increases rapidly towards infinity. Therefore, \[ \lim_{x \to 3} g(x) \to \infty \] - **For \( \lim_{x \to 1} g(x) \)**: Observing the graph, as \( x \) approaches \( 1 \), \( g(x) \) appears to be around \( 6 \). Therefore, \[ \lim_{x \to 1} g(x) \approx 6 \] - **For \( \lim_{x \to \infty} g(x) \)**: Observing the graph, as \( x \) continues to increase towards positive infinity, \( g(x) \) decreases towards \( 0 \). Therefore, \[ \lim_{x \to \infty} g(x) = 0 \]
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