Chapter 3: Applications of Differentiation
Section 3.9: Indeterminate Forms and L'Hôpital's Rule
Evaluate limx→0+x lnx, then detail an applicable strategy taken from Table 3.9.1.
As x→0 from the right, lnx→−∞, so the product x lnx tends to the indeterminate form 0⋅∞. This form is converted to either 0/0 or ∞/∞ by algebraically rewriting the product as a quotient, then applying L'Hôpital's rule. The full solution is as follows.
If the product is written as the quotient x1lnx, L'Hôpital's rule would apply, but it would be very difficult to obtain the limit of the resulting expression.
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→0+x lnx→show solution stepsLimit Stepslimx→0+⁡x⁢ln⁡x▫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 appliedx⁢ln⁡x=−xThis gives:limx→0+⁡−x▫2. Apply the constant multiple rule to the term limx→0⁡−x◦Recall the definition of the constant multiple rulelimx→0⁡⁢f⁡x=⁢limx→0⁡f⁡x◦This means:limx→0⁡−x=We can rewrite the limit as:−limx→0+⁡x▫3. Apply the identity rule◦Recall the definition of the identity rulelimx→a⁡x=aThis gives:0
Alternatively, see Tables 3.9.7(a - d).
tutor to launch the Limit Methods tutor for this example. See Figure 3.9.7(a) for the relevant initial state of the tutor. Note the selection of "right" as the Direction.
Figure 3.9.7(a) Limit Methods tutor applied to limx→0+x lnx
Click the L'Hôpital's rule button to rewrite x lnx as lnx1/x. As per Figure 3.9.7(b), select x under the "Numerator:" heading, and click the right-facing arrow. (Note the "Click here" reminder in the figure.) The result is seen in Figure 3.9.7(c) where lnx is listed as the numerator, and 1/x is listed as the denominator. The display shows the limit in a form for which L'Hôpital's rule applies.
Figure 3.9.7(b) Dialog for L'Hôpital's rule
Figure 3.9.7(c) Dialog for L'Hôpital's rule
Click the "Apply" button to obtain the configuration in Figure 3.9.7(d).
Figure 3.9.7(d) State of the Limit Methods tutor after L'Hôpital's rule has been applied
All that remains is to apply the Constant Multiple and Identity rules to obtain limx→0+x lnx=0.
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