Chapter 3: Applications of Differentiation
Section 3.9: Indeterminate Forms and L'Hôpital's Rule
Evaluate limx→0sinxx, then detail an applicable strategy taken from Table 3.9.1.
Evaluate the given limit
Context Panel: Evaluate and Display Inline
limx→0sinxx = 1
Apply L'Hôpital's rule
The application of L'Hôpital's rule is valid because as x→0, the fraction sinx/x tends to the indeterminate form 0/0.
This limit is easily obtained by L'Hôpital's rule. However, to apply this rule, the derivative of sinx must already be known. The derivation of this derivative is given in Table 2.7.5, which in turn invokes the special trig limits in Table 1.4.1. On a logical basis, these earlier calculations were essential, and could not have been deleted in favor of L'Hôpital's rule without tearing a hole in the logical structure of the calculus.
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→0sinxx→show solution stepsLimit Stepslimx→0⁡sin⁡xx▫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⁡xx=cos⁡xThis gives:limx→0⁡cos⁡x▫2. Evaluate the limit of cos(x)◦Recall the definition of the cos rulelimx→a⁡cos⁡f⁡x=cos⁡limx→a⁡f⁡xThis gives:1
This stepwise solution can be worked out interactively via the Context Panel or the
tutor. The appropriate rule to apply is, of course, L'Hôpital's rule.
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