The figure below shows the graph of a quadratic function, f(x), whose limit at an unspecified x-coordinate evaluates to L (shown as a red point on the y-axis). A positive value for & has been chosen. Complete the following tasks: 1. Adjust the value of a by sliding the purple movable point to obtain the approximate x-coordinate at which the limit of the function evaluates to L. In other words, find the value of a such that limf(x) = L. x→a Assume that a > 0 for this quadratic function. 2. Adjust the value of 8 by sliding the orange movable points to obtain an approximate interval of x such that if 0 < xa| < 6, then f(x) - L < &. Note that this assessment does not require you to obtain exact measures of a and 8 since the exact form of f(x) is not given. Instead, you will use the graph to approximate a and 6 in a way that agrees with your understanding of the formal epsilon-delta definition of the limit. Provide your answer below: -10 RESET -5 10 L + ε L L -5 0 -5 -10 a a + d 5 10
The figure below shows the graph of a quadratic function, f(x), whose limit at an unspecified x-coordinate evaluates to L (shown as a red point on the y-axis). A positive value for & has been chosen. Complete the following tasks: 1. Adjust the value of a by sliding the purple movable point to obtain the approximate x-coordinate at which the limit of the function evaluates to L. In other words, find the value of a such that limf(x) = L. x→a Assume that a > 0 for this quadratic function. 2. Adjust the value of 8 by sliding the orange movable points to obtain an approximate interval of x such that if 0 < xa| < 6, then f(x) - L < &. Note that this assessment does not require you to obtain exact measures of a and 8 since the exact form of f(x) is not given. Instead, you will use the graph to approximate a and 6 in a way that agrees with your understanding of the formal epsilon-delta definition of the limit. Provide your answer below: -10 RESET -5 10 L + ε L L -5 0 -5 -10 a a + d 5 10
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Understanding Limits through Graphical Adjustments
The figure below shows the graph of a quadratic function, \( f(x) \), whose limit at an unspecified \( x \)-coordinate evaluates to \( L \) (shown as a red point on the \( y \)-axis). A positive value for \( \epsilon \) has been chosen. Complete the following tasks:
1. **Adjust the value of \( a \) by sliding the purple movable point to obtain the approximate \( x \)-coordinate at which the limit of the function evaluates to \( L \).** In other words, find the value of \( a \) such that
\[
\lim_{{x \to a}} f(x) = L.
\]
Assume that \( a > 0 \) for this quadratic function.
2. **Adjust the value of \( \delta \) by sliding the orange movable points** to obtain an approximate interval of \( x \) such that if \( 0 < |x - a| < \delta \), then \( |f(x) - L| < \epsilon \).
Note that this assessment does not require you to obtain exact measures of \( a \) and \( \delta \) since the exact form of \( f(x) \) is not given. Instead, you will use the graph to approximate \( a \) and \( \delta \) in a way that agrees with your understanding of the formal epsilon-delta definition of the limit.
#### Provide your answer below:
[Graph Details]
- The graph features a U-shaped quadratic function \( f(x) \).
- The red point on the \( y \)-axis indicates the limit \( L \).
- Two horizontal dashed lines labeled \( L + \epsilon \) and \( L - \epsilon \) create a vertical band around \( L \).
- A purple movable point on the graph can be adjusted to find the \( x \)-coordinate \( a \) where \( \lim_{{x \to a}} f(x) = L \).
- Orange movable points on the \( x \)-axis labeled \( a - \delta \) and \( a + \delta \) determine the interval around \( a \) for which the condition \( 0 < |x - a| < \delta \) holds true.
- The green shaded region between \( L + \epsilon \) and \( L - \epsilon \) on the \( y \)-axis](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fffcf92b5-4154-452a-af6b-b4e696e552b0%2F21b30289-f742-4a0e-b2d8-d2bf03f4f8ae%2Fs7ya44_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Limits through Graphical Adjustments
The figure below shows the graph of a quadratic function, \( f(x) \), whose limit at an unspecified \( x \)-coordinate evaluates to \( L \) (shown as a red point on the \( y \)-axis). A positive value for \( \epsilon \) has been chosen. Complete the following tasks:
1. **Adjust the value of \( a \) by sliding the purple movable point to obtain the approximate \( x \)-coordinate at which the limit of the function evaluates to \( L \).** In other words, find the value of \( a \) such that
\[
\lim_{{x \to a}} f(x) = L.
\]
Assume that \( a > 0 \) for this quadratic function.
2. **Adjust the value of \( \delta \) by sliding the orange movable points** to obtain an approximate interval of \( x \) such that if \( 0 < |x - a| < \delta \), then \( |f(x) - L| < \epsilon \).
Note that this assessment does not require you to obtain exact measures of \( a \) and \( \delta \) since the exact form of \( f(x) \) is not given. Instead, you will use the graph to approximate \( a \) and \( \delta \) in a way that agrees with your understanding of the formal epsilon-delta definition of the limit.
#### Provide your answer below:
[Graph Details]
- The graph features a U-shaped quadratic function \( f(x) \).
- The red point on the \( y \)-axis indicates the limit \( L \).
- Two horizontal dashed lines labeled \( L + \epsilon \) and \( L - \epsilon \) create a vertical band around \( L \).
- A purple movable point on the graph can be adjusted to find the \( x \)-coordinate \( a \) where \( \lim_{{x \to a}} f(x) = L \).
- Orange movable points on the \( x \)-axis labeled \( a - \delta \) and \( a + \delta \) determine the interval around \( a \) for which the condition \( 0 < |x - a| < \delta \) holds true.
- The green shaded region between \( L + \epsilon \) and \( L - \epsilon \) on the \( y \)-axis
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