Chapter 1: Limits
Section 1.3: Limit Laws
Example 1.3.2
Use the laws in Table 1.3.1 to evaluate limx→1x2−3 x+6.
Solution
Table 1.3.2(a) summarizes the application of rules taken from Table 1.3.1.
Result of Rule Application
Rule Applied
limx→1x2−3 x+6
=limx→1x2−limx→13x+limx→1 6
difference and sum rules
=limx→1x2−3limx→1x+limx→16
power and constant-multiple rules
=12−3 1+6
identity and constant rules applied
=4
basic arithmetic ⇒ final answer
Table 1.3.2(a) Stepwise evaluation of limx→1x2−3 x+6
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
Loading Student:-Calculus1
limx→1x2−3 x+6→show solution stepsLimit Stepslimx→1x2−3x+6▫1. Apply the sum rule◦Recall the definition of the sum rulelimx→af1x+f2x+f3x=limx→af1x+limx→af2x+limx→af3xf1x=x2f2x=−3xf3x=6This gives:limx→1x2+limx→1−3x+limx→16▫2. Apply the constant rule to the term limx→16◦Recall the definition of the constant ruleLimitC,x=C◦This meanslimx→16=6We can now rewrite the limit as:limx→1x2+limx→1−3x+6▫3. Apply the constant multiple rule to the term limx→1−3x◦Recall the definition of the constant multiple rulelimx→1Cfx=Climx→1fx◦This means:limx→1−3x=−3⋅limx→1xWe can rewrite the limit as:limx→1x2−3limx→1x+6▫4. Apply the identity rule◦Recall the definition of the identity rulelimx→ax=aThis gives:limx→1x2+3▫5. Apply the power rule◦Recall the definition of the power rulelimx→axn=limx→axnThis gives:limx→1x2+3▫6. Apply the identity rule◦Recall the definition of the identity rulelimx→ax=aThis gives:4
Table 1.3.2(b) Maple's stepwise solution via the All Solution Steps option in the Context Panel
Finally, to evaluate the given limit interactively with the Limit Methods tutor, press the following button.
A tutor can be launched from the Tools≻Tutors menu, or from the Context Panel after the appropriate package has been loaded.
To specify a problem in the Limit Methods tutor, note that the top line of this maplet contains fields for the function, variable, limit point, and whether the limit is two-sided (blank) or one-sided (left or right). Press the Start button, then apply limit laws by clicking the corresponding button in the tutor. The menu bars provide a summary of each known rule (Rule Definition), help, and another way to apply rules (Apply the Rule). Note that the selected rule is generally applied to the first possible occurrence; it may be necessary to apply a rule multiple times in succession.
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