Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![**Problem Statement for Educational Website:**
For the function \( f(x) \) shown below, determine \( \lim_{{x \to -2^-}} f(x) \).
**Graph Description:**
The provided graph depicts a function \( f(x) \) on a Cartesian coordinate plane. The horizontal axis is labeled as \( x \) and extends from -10 to 10, and the vertical axis is labeled as \( y \) and extends from -10 to 10.
**Detailed Analysis of the Graph:**
- The function \( f(x) \) is represented by two distinct parts due to discontinuities at \( x = -2 \) and \( x = 0 \).
1. **Left-side Behavior (\( x < -2 \))**:
- The graph shows a line segment that extends downward and to the left, intersecting the point (-2, 6) with an open circle, indicating that the value at \( x = -2 \) is not included at this point of the segment.
2. **Right-side Behavior (\( x > -2 \))**:
- The graph shows another line segment that starts at (-2, -4) with a closed circle, indicating that the value at \( x = -2 \) is included. This segment extends down and to the left.
**Evaluation of \( \lim_{{x \to -2^-}} f(x) \)**:
To evaluate the limit from the left as \( x \) approaches -2, we observe the behavior of \( f(x) \) as \( x \) gets closer to -2 from the left-hand side:
- As \( x \to -2 \) from the left, the function values approach 6.
Therefore, the left-hand limit of \( f(x) \) as \( x \) approaches -2 is:
\[ \lim_{{x \to -2^-}} f(x) = 6 \]
Thus, the answer is:
\[ \lim_{{x \to -2^-}} f(x) = 6 \]](https://content.bartleby.com/qna-images/question/991e22e9-7073-4035-a9d1-c12914250c1e/e4331ed8-f925-42c1-b90e-1e5f2e6358f6/eoq2p9n_processed.png)
Transcribed Image Text:**Problem Statement for Educational Website:**
For the function \( f(x) \) shown below, determine \( \lim_{{x \to -2^-}} f(x) \).
**Graph Description:**
The provided graph depicts a function \( f(x) \) on a Cartesian coordinate plane. The horizontal axis is labeled as \( x \) and extends from -10 to 10, and the vertical axis is labeled as \( y \) and extends from -10 to 10.
**Detailed Analysis of the Graph:**
- The function \( f(x) \) is represented by two distinct parts due to discontinuities at \( x = -2 \) and \( x = 0 \).
1. **Left-side Behavior (\( x < -2 \))**:
- The graph shows a line segment that extends downward and to the left, intersecting the point (-2, 6) with an open circle, indicating that the value at \( x = -2 \) is not included at this point of the segment.
2. **Right-side Behavior (\( x > -2 \))**:
- The graph shows another line segment that starts at (-2, -4) with a closed circle, indicating that the value at \( x = -2 \) is included. This segment extends down and to the left.
**Evaluation of \( \lim_{{x \to -2^-}} f(x) \)**:
To evaluate the limit from the left as \( x \) approaches -2, we observe the behavior of \( f(x) \) as \( x \) gets closer to -2 from the left-hand side:
- As \( x \to -2 \) from the left, the function values approach 6.
Therefore, the left-hand limit of \( f(x) \) as \( x \) approaches -2 is:
\[ \lim_{{x \to -2^-}} f(x) = 6 \]
Thus, the answer is:
\[ \lim_{{x \to -2^-}} f(x) = 6 \]
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