Chapter 1: Limits
Section 1.3: Limit Laws
In general, limits can be evaluated by taking the limit of individual terms within an expression and then combining these results using the appropriate mathematical operations. For example, the limit of a sum is the sum of the limits. Table 1.3.1 summarizes the fundamental properties of limits.
limx→a fx+gx=limx→a fx+limx→ag⁡x
limx→a⁡fx and limx→ag⁡x exist
limx→a⁡fx and limx→ag⁡x exist, and limx→ag⁡x≠⁢0
limx→a fx1/n=limx→a fx1/n
limx→a⁡fx exists, and if n is even, limx→a⁡fx⁢>⁢0
Table 1.3.1 Fundamental properties of limits
Note the conditions listed in the third column. If the conditions for a rule are not satisfied, the rule cannot be used to justify the evaluation of a limit.
Maple Tools for Limits
In addition to the limit operator in the Calculus palette, and the limit option in the Context Panel, Maple has the Limit Methods tutor, an interactive tool for implementing the limit laws in Table 1.3.1. The tutor can be launched from the Tools≻Tutors menu, and is found in the section "Calculus - Single Variable." If launched from this menu, the limit problem must be entered into the tutor.
Alternatively, with the Student Calculus1 package loaded, if the limit operator from the Calculus palette is used to pose the limit problem, then the Limit Methods tutor can be launched from the Context Panel, and the limit problem will be pre-installed in the tutor.
Finally, with the Student Calculus1 package loaded, the Student Calculus1≻All Solution Steps option in the Context Panel for a limit problem provides a complete annotated stepwise solution of the limit problem. Additionally, the Context Panel provides access to the rules for evaluating limits, as per the following figure.
The Limit Methods Tutor
Figure 1.3.1 contains a screen-shot of the
tutor applied to the calculation of limx→1x2−1x−1.
Figure 1.3.1 The Limit Methods tutor applied to limx→1x2−1x−1
The following rules were applied: Factor, Sum, Identity, and Constant. The Factor rule does not appear in Table 1.3.1, and neither does l'Hôpital's Rule or the Change, Divide by Zero, and Rewrite rules. The Change rule is used to change variables in a limit calculation, whereas the Rewrite rule is used to rewrite the form of an expression. The Divide by Zero rule and l'Hôpital's rule will be used in conjunction with more advanced limit calculations later in the course. (See the note in Section 3.9 on the spelling and pronunciation of the French name L'Hôpital. The astute reader will observe that the spelling in the tutor differs from the spelling adopted in this text.)
The Understood Rules button at the top of the tutor can be used to signal Maple that a particular rule is "understood" and does not need to be explicitly invoked. Maple will then automatically apply such "understood" rules.
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.
When the Close button on the Limit Methods tutor is pressed, an annotated solution such as the one in Table 1.3.2 is returned to the worksheet.
Table 1.3.2 Annotated solution returned by the
Note that the tutor can suggest steps that are not optimal. It is a tool for experimentation, not an infallible oracle.
The "All Solution Steps" Option
An annotated stepwise solution such as the one in Table 1.3.3 is obtained through the Context Panel.
Tools≻Load Package: Student Calculus 1
limx→1x2−1x−1→show solution stepslimx→1⁡x2−1x−1=limx→1⁡x+1factor=limx→1⁡x+limx→1⁡1sum=limx→1⁡x+1constant=2identity
Table 1.3.3 All Solution Steps option from the Context Panel applied to a limit calculation
The highlighted portion of Table 1.3.3 is the return of the Context Panel option. The inert limit operator at the extreme left, and the arrow extending from it are what cause the highlighted material to appear.
The path taken through the calculation by the tutor, and the path taken by the option "Student Calculus1≻All Solution Steps" option can differ.
The Squeeze Theorem
The Squeeze (or Sandwich) Theorem
The functions f, g, and h satisfy fx≤gx≤hx for all x near a (except possibly at x=a)
limx→a fx=L and limx→a hx=L
Thus, if the values of gx are bounded above and below by quantities that themselves tend to a limit L, then the values of gx must themselves tend towards L.
The Limit laws, including the Squeeze theorem, provide rigorous tools for evaluating limits without having to provide ϵ-δ proofs. The essence of this chapter is found in Table 1.3.1, a compendium of the rules by which limits are actually evaluated in practice.
Use the laws in Table 1.3.1 to evaluate limx→32 x−4.
Use the laws in Table 1.3.1 to evaluate limx→1x2−3 x+6.
Use the laws in Table 1.3.1 to evaluate limt→−2t2+5⁢t⁢t10−6⁢t5+2⁢tt4+4⁢t2+4.
Use the Squeeze theorem to show limx→0x2 sin1/x=0.
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