Chapter 3: Applications of Differentiation
Section 3.9: Indeterminate Forms and L'Hôpital's Rule
Evaluate limx→∞exx2, then detail an applicable strategy taken from Table 3.9.1.
Evaluation of the Limit
Write limx→∞…, being sure to use the exponential "e"
Context Panel: Evaluate and Display Inline
limx→∞⁡ⅇxx2 = ∞
The exponential "e" 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 "e". 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 limx→∞…, again being sure to use the exponential "e".
Context Panel: Student Calculus1≻All Solution Steps
limx→∞ⅇxx2→show solution stepsLimit Stepslimx→∞⁡ⅇxx2▫1. Apply the L'Hôpital's Rule rule◦Recall the definition of the L'Hôpital's Rule rulelimx→c⁡f⁡xg⁡x=limx→c⁡ⅆⅆxf⁡xⅆⅆxg⁡x◦Rule appliedⅇxx2=ⅇx2⁢xThis gives:limx→∞⁡ⅇx2⁢x▫2. Apply the constant multiple rule to the term limx→∞⁡ⅇx2⁢x◦Recall the definition of the constant multiple rulelimx→∞⁡⁢f⁡x=⁢limx→∞⁡f⁡x◦This means:limx→∞⁡ⅇx2⁢x=We can rewrite the limit as:limx→∞⁡ⅇxx2▫3. Apply the L'Hôpital's Rule rule◦Recall the definition of the L'Hôpital's Rule rulelimx→c⁡f⁡xg⁡x=limx→c⁡ⅆⅆxf⁡xⅆⅆxg⁡x◦Rule appliedⅇxx=ⅇxThis gives:limx→∞⁡ⅇx2▫4. Apply the exponential rule◦Recall the definition of the exponential rulelimx→a⁡ef⁡x=elimx→a⁡f⁡xThis gives:∞
L'Hôpital's rule is valid because ex/x2 tends to the indeterminate form ∞/∞ as x→∞. Maple then carefully moves the multiplicative 2 from the denominator and again applies L'Hôpital's rule because the fraction ex/x again tends to the indeterminate form ∞/∞. After the second application of L'Hôpital's rule, the fraction has become ex/2, which has the immediate limit of ∞. (Maple obtains the limit of ex 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.
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