The graph of f(x) is shown. 9r f(z) = Not dran o cale (a) Find the following limits. L- lim f(x) = K- lim f(x) =
The graph of f(x) is shown. 9r f(z) = Not dran o cale (a) Find the following limits. L- lim f(x) = K- lim f(x) =
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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![### The graph of f(x) is shown.
\[ f(x) = \frac{9x}{\sqrt{x^2 + 5}} \]
- The graph depicts the function \( f(x) = \frac{9x}{\sqrt{x^2 + 5}} \).
- It is annotated with two dashed red lines, presumably horizontal asymptotes, labeled \( L \) and \( K \).
The graph shows:
- A curve approaching the x-axis from below as \( x \) decreases without bound (indicative of a horizontal asymptote at \( K \)).
- The curve crosses the y-axis, indicating the value of \( f(0) \).
- As \( x \) increases, the curve approaches another horizontal line, \( L \).
#### (a) Find the following limits.
\[ L = \lim_{{x \to +\infty}} f(x) = \boxed{} \]
\[ K = \lim_{{x \to -\infty}} f(x) = \boxed{} \]
#### (b) Determine \( x_1 \) and \( x_2 \) in terms of \( \varepsilon \).
\[ x_1 = \boxed{} \]
\[ x_2 = \boxed{} \]
#### (c) Determine \( M \), where \( M > 0 \), such that \( |f(x) - L| < \varepsilon \) for \( x > M \).
\[ M = \boxed{} \]
#### (d) Determine \( N \), where \( N < 0 \), such that \( |f(x) - K| < \varepsilon \) for \( x < N \).
\[ N = \boxed{} \]
### Explanation of the Graph
- **Function:** The function \( f(x) = \frac{9x}{\sqrt{x^2 + 5}} \) is illustrated by the curve.
- **Asymptotes:** The red dashed lines delineate horizontal asymptotes. The function approaches these lines as \( x \to \pm \infty \).
- **Plot Details:** The curve approximates these horizontal lines but never actually intersects them.
### Instructions for Students
- **Limits:** Evaluate the horizontal limits by analyzing the behavior of \( f(x) \) as \( x \) approaches positive and negative infinity.
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F08c083ee-5dc4-4392-bfaa-a5f4117ce6af%2F00079470-34dc-4c1a-9600-0a9d1a77597c%2Fcueail.jpeg&w=3840&q=75)
Transcribed Image Text:### The graph of f(x) is shown.
\[ f(x) = \frac{9x}{\sqrt{x^2 + 5}} \]
- The graph depicts the function \( f(x) = \frac{9x}{\sqrt{x^2 + 5}} \).
- It is annotated with two dashed red lines, presumably horizontal asymptotes, labeled \( L \) and \( K \).
The graph shows:
- A curve approaching the x-axis from below as \( x \) decreases without bound (indicative of a horizontal asymptote at \( K \)).
- The curve crosses the y-axis, indicating the value of \( f(0) \).
- As \( x \) increases, the curve approaches another horizontal line, \( L \).
#### (a) Find the following limits.
\[ L = \lim_{{x \to +\infty}} f(x) = \boxed{} \]
\[ K = \lim_{{x \to -\infty}} f(x) = \boxed{} \]
#### (b) Determine \( x_1 \) and \( x_2 \) in terms of \( \varepsilon \).
\[ x_1 = \boxed{} \]
\[ x_2 = \boxed{} \]
#### (c) Determine \( M \), where \( M > 0 \), such that \( |f(x) - L| < \varepsilon \) for \( x > M \).
\[ M = \boxed{} \]
#### (d) Determine \( N \), where \( N < 0 \), such that \( |f(x) - K| < \varepsilon \) for \( x < N \).
\[ N = \boxed{} \]
### Explanation of the Graph
- **Function:** The function \( f(x) = \frac{9x}{\sqrt{x^2 + 5}} \) is illustrated by the curve.
- **Asymptotes:** The red dashed lines delineate horizontal asymptotes. The function approaches these lines as \( x \to \pm \infty \).
- **Plot Details:** The curve approximates these horizontal lines but never actually intersects them.
### Instructions for Students
- **Limits:** Evaluate the horizontal limits by analyzing the behavior of \( f(x) \) as \( x \) approaches positive and negative infinity.
-
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