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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Hi, thanks for answering this question. I have another question about this. For the part circled in green, where did this come from? Is there a name for this rule? I never saw this anywhere in my notes so it's throwing me off.
![The image contains a mathematical expression for educational purposes. The expression is written as follows:
36.
\[ \left\{ \frac{e^{n/10}}{2^n} \right\} \]
There are small, hand-drawn shapes above and beside the number "36," resembling a loop or arc and a small circle, but they do not appear to relate directly to the mathematical expression.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61b75115-d70f-4fe2-af93-2076876ad69a%2F5d7efdb7-7db9-42bf-a882-50296bf2a7bd%2Fzzsmrnp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains a mathematical expression for educational purposes. The expression is written as follows:
36.
\[ \left\{ \frac{e^{n/10}}{2^n} \right\} \]
There are small, hand-drawn shapes above and beside the number "36," resembling a loop or arc and a small circle, but they do not appear to relate directly to the mathematical expression.
![## Calculus Example: Exponential Limit
### Step 1
The given data is:
\[
e^{1/10} = \sqrt[10]{e} = 1.1
\]
\[
r = \frac{e^{1/10}}{2}, \quad 0 < r < 1
\]
To find the value of:
\[
\lim_{{n \to \infty}} \frac{e^{n/10}}{2^n}
\]
### Step 2
Use the formula:
\[
\lim_{{n \to \infty}} a^n = \infty, \quad \text{if } a > 1
\]
\[
\lim_{{n \to \infty}} a^n = 0, \quad \text{if } 0 < a < 1
\]
Now,
\[
\lim_{{n \to \infty}} \frac{e^{n/10}}{2^n} = \lim_{{n \to \infty}} r^n
\]
\[
= 0 \quad (\because 0 < r < 1)
\]
This example demonstrates the application of limits to exponential functions, useful in evaluating the behavior of functions as \( n \) approaches infinity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61b75115-d70f-4fe2-af93-2076876ad69a%2F5d7efdb7-7db9-42bf-a882-50296bf2a7bd%2Fxb582a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Calculus Example: Exponential Limit
### Step 1
The given data is:
\[
e^{1/10} = \sqrt[10]{e} = 1.1
\]
\[
r = \frac{e^{1/10}}{2}, \quad 0 < r < 1
\]
To find the value of:
\[
\lim_{{n \to \infty}} \frac{e^{n/10}}{2^n}
\]
### Step 2
Use the formula:
\[
\lim_{{n \to \infty}} a^n = \infty, \quad \text{if } a > 1
\]
\[
\lim_{{n \to \infty}} a^n = 0, \quad \text{if } 0 < a < 1
\]
Now,
\[
\lim_{{n \to \infty}} \frac{e^{n/10}}{2^n} = \lim_{{n \to \infty}} r^n
\]
\[
= 0 \quad (\because 0 < r < 1)
\]
This example demonstrates the application of limits to exponential functions, useful in evaluating the behavior of functions as \( n \) approaches infinity.
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