Chapter 3: Applications of Differentiation
Section 3.9: Indeterminate Forms and L'Hôpital's Rule
|
Example 3.9.5
|
|
Evaluate , then detail an applicable strategy taken from Table 3.9.1.
|
|
|
|
Solution
|
|
|
Evaluation of the Limit
|
|
•
|
Write , being sure to use the exponential ""
Context Panel: Evaluate and Display Inline
|
|
=
|
|
|
The exponential "" is available in the Common Symbols and Constants and Symbols palettes. Alternatively, typing the letter "e" and using Command Completion brings up a list of Maple entries starting with that letter, including the exponential "". Command Completion is available from the Tools menu, or by pressing the Escape key.
|
|
Annotated Stepwise Maple Solution
|
|
•
|
The Context Panel provides access to the options All Solutions, Next Step, and Limit Rules, as per the figure to the right.
|
•
|
While the Context Panel lists both a "difference" and a "sum" rule, Maple treats a difference as if it were a sum, thereby making no essential distinction between the two rules. Thus, where Table 1.3.1 distinguishes between a Sum and a Difference rule, Maple considers both to be the single Sum rule.
|
•
|
In addition, the Student Calculus1 package contains a ShowSolution command that can be applied to the inert form of the limit operator. The limit operator can be converted to the inert form through the Context Panel by selecting the options 2-D Math≻Convert To≻Inert Form.
|
|
|
|
|
Annotated stepwise solution via the Context Panel
|
•
|
Tools≻Load Package: Student Calculus 1
|
•
|
Write , again being sure to use the exponential "".
|
•
|
Context Panel: Student Calculus1≻All Solution Steps
|
|
Loading Student:-Calculus1
|
|
|
|
L'Hôpital's rule is valid because tends to the indeterminate form as . Maple then carefully moves the multiplicative 2 from the denominator and again applies L'Hôpital's rule because the fraction again tends to the indeterminate form . After the second application of L'Hôpital's rule, the fraction has become , which has the immediate limit of ∞. (Maple obtains the limit of by application of the Exponential rule, which says that the limit of the exponential is the exponential of the limit.)
A similar solution can be obtained with the
tutor, provided either the "All Steps" button is pressed, or the L'Hôpital's rule button is used to apply that rule.
|
|
|
<< Previous Example Section 3.9
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
|