Chapter 3: Applications of Differentiation
Section 3.9: Indeterminate Forms and L'Hôpital's Rule
Evaluate limx→0cscx−cotx, then detail an applicable strategy taken from Table 3.9.1.
Figure 3.9.8(a) contains graphs of cscx (in black) and cotx (in red). The line x=0 is a vertical asymptote for both curves, suggesting that as x→0, the difference cscx−cotx tends to the indeterminate form ∞−∞.
Figure 3.9.8(a) also suggests that the values of the functions near x=0 are so similar that perhaps the limit of the difference cscx−cotx might indeed be zero. Of course, this can be verified immediately in Maple via the calculation
limx→0cscx−cotx = 0
Figure 3.9.8(a) cscx (black) and cotx (red)
The difference cscx−cotx needs to be rewritten so that as x→0, it tends to the indeterminate form 0/0 or ∞/∞. The recipe in Table 3.9.2 leads to the same result as that obtained by basic trigonometry, namely,
As x→0, this latter tends to the indeterminate form 0/0, which yields to L'Hôpital's rule. The complete calculation would then be
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→0cscx−cotx→show solution stepsLimit Stepslimx→0⁡csc⁡x−cot⁡x▫1. Rewrite◦Equivalent expressioncsc⁡x−cot⁡x=−−1+cos⁡xsin⁡xThis gives:limx→0⁡−−1+cos⁡xsin⁡x▫2. Apply the constant multiple rule to the term limx→0⁡−−1+cos⁡xsin⁡x◦Recall the definition of the constant multiple rulelimx→0⁡⁢f⁡x=⁢limx→0⁡f⁡x◦This means:limx→0⁡−−1+cos⁡xsin⁡x=We can rewrite the limit as:−limx→0⁡−1+cos⁡xsin⁡x▫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 applied−1+cos⁡xsin⁡x=−sin⁡xcos⁡xThis gives:−limx→0⁡−sin⁡xcos⁡x▫4. Apply the constant multiple rule to the term limx→0⁡−sin⁡xcos⁡x◦Recall the definition of the constant multiple rulelimx→0⁡⁢f⁡x=⁢limx→0⁡f⁡x◦This means:limx→0⁡−sin⁡xcos⁡x=We can rewrite the limit as:limx→0⁡sin⁡xcos⁡x▫5. Apply the quotient rule◦Recall the definition of the quotient rulelimx→a⁡f⁡xg⁡x=limx→a⁡f⁡xlimx→a⁡g⁡xf⁡x=sin⁡xg⁡x=cos⁡xThis gives:limx→0⁡sin⁡xlimx→0⁡cos⁡x▫6. Evaluate the limit of sin(x)◦Recall the definition of the sin rulelimx→a⁡sin⁡f⁡x=sin⁡limx→a⁡f⁡xThis gives:0
This solution, or a shorter one, can be obtained with the
tutor. Selecting, for example, the Constant Multiple rule in the menu "Understood Rules" will suppress the step in which that rule is explicitly invoked.
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