Chapter 3: Applications of Differentiation
Section 3.9: Indeterminate Forms and L'Hôpital's Rule
Evaluate limx→∞2 x2−13 x2+5 x, then detail an applicable strategy taken from Table 3.9.1.
Evaluation of the Limit
Context Panel: Evaluate and Display Inline
limx→∞2 x2−13 x2+5 x = 23
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
Calculus palette: Limit template
Context Panel: Student Calculus1≻All Solution Steps
limx→∞2 x2−13 x2+5 x→show solution stepsLimit Stepslimx→∞⁡2⁢x2−13⁢x2+5⁢x▫1. Factor◦Factor a polynomial or rational function2⁢x2−13⁢x2+5⁢x=2⁢x2−1x⁢3⁢x+5This gives:limx→∞⁡2⁢x2−1x⁢3⁢x+5▫2. 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 applied2⁢x2−1x⁢3⁢x+5=4⁢x6⁢x+5This gives:limx→∞⁡4⁢x6⁢x+5▫3. Apply the constant multiple rule to the term limx→∞⁡4⁢x6⁢x+5◦Recall the definition of the constant multiple rulelimx→∞⁡⁢f⁡x=⁢limx→∞⁡f⁡x◦This means:limx→∞⁡4⁢x6⁢x+5=We can rewrite the limit as:4⁢limx→∞⁡x6⁢x+5▫4. 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 appliedx6⁢x+5=16This gives:4⁢limx→∞⁡16▫5. Apply the constant rule to the term limx→∞⁡16◦Recall the definition of the constant ruleLimit⁡C,x=C◦This meanslimx→∞⁡16=16We can now rewrite the limit as:23
Alternatively, see Table 3.9.6(a).
Table 3.9.6(a) Solution by Limit Methods tutor
This solution is obtained interactively via the
tutor. The content of Table 3.9.6(a) is obtained by twice selecting L'Hôpital's rule as the applicable rule. After obtaining the complete solution, clicking the Close button in the tutor returns the content of Table 3.9.6(a).
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