Chapter 3: Applications of Differentiation
Section 3.9: Indeterminate Forms and L'Hôpital's Rule
Evaluate limx→0sin2xx, then detail an applicable strategy taken from Table 3.9.1.
Evaluation of the Limit
Context Panel: Evaluate and Display Inline
limx→0sin2xx = 0
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
Context Panel: Student Calculus1≻All Solution Steps
limx→0sin2xx→show solution stepsLimit Stepslimx→0⁡sin⁡x2x▫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 appliedsin⁡x2x=2⁢sin⁡x⁢cos⁡xThis gives:limx→0⁡2⁢sin⁡x⁢cos⁡x▫2. Apply the constant multiple rule to the term limx→0⁡2⁢sin⁡x⁢cos⁡x◦Recall the definition of the constant multiple rulelimx→0⁡⁢f⁡x=⁢limx→0⁡f⁡x◦This means:limx→0⁡2⁢sin⁡x⁢cos⁡x=We can rewrite the limit as:2⁢limx→0⁡sin⁡x⁢cos⁡x▫3. Apply the product rule◦Recall the definition of the product rulelimx→a⁡f⁡x⁢g⁡x=limx→a⁡f⁡x⁢limx→a⁡g⁡xf⁡x=sin⁡xg⁡x=cos⁡xThis gives:2⁢limx→0⁡sin⁡x⁢limx→0⁡cos⁡x▫4. Evaluate the limit of sin(x)◦Recall the definition of the sin rulelimx→a⁡sin⁡f⁡x=sin⁡limx→a⁡f⁡xThis gives:0
The help page Limit Rules states that the Product rule will not split an indeterminate form. The implication is that Maple first checks the limits of the factors and only applies the rule if such is valid. That is why immediately after the application of the Product rule Maple applies the sine rule and gets zero for the limit: Maple already knows that the limit of the other factor is finite.
Alternate Stepwise Solution
Table 3.9.2(a) Annotated stepwise solution using the Limit Methods tutor
This solution is obtained interactively via the
tutor. The content of Table 3.9.2(a) is obtained by 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 3.9.2(a).
After Maple has applied the Product rule, the solution in Table 3.9.2(a) invokes the cosine rule to verify that that limit is finite.
Of course, the use of L'Hôpital's rule is justified because as x→0, the fraction sin2x/x tends to the indeterminate form 0/0.
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