LimitLaws.mws

Limit Laws

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Lesson Overview

If all limits had to be evaluated using the epsilon-delta  definition, limits would be extremely tedious to use. Fortunately, there are some general rules, called the Limit Laws, for evaluating limits that allow us to avoid proving each and every limit that we encounter.

The most important skill to develop during this lesson is the ability to evaluate a limit by applying the Limit Laws in an appropriate order.

The Limit  maplet [ Maplet Viewer] [ MapleNet] provides an excellent tool to learn the names of the limit laws and how they can be applied to evaluate a limit. This is demonstrated with three examples.

The Squeeze Theorem is a limit law that is somewhat different from all of the other laws. It depends on knowing information about two related functions. Because of the different nature of this result, it is presented separate from the other laws. One example of its application is given in this lesson; additional examples are given in the Limits of Trigonometric Functions lesson

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The 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 appropirate mathematical operations. For example, the limit of a sum is the sum of the limits. The following table summarizes the fundamental properties of limits.

  MATRIX([[Name, Formula, `Extra Conditions`], [___________________, _______________________________________________, ____________________________], [Constant, limit(c,x = a) = c, none], [Identity, limit...  

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.

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Three Examples

The following three examples demonstrate the use of these limit laws in the evaluation of limits. The first two examples are the first two examples considered in Lesson 2; the final example is new.

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Example 1: Use the Limit Laws to evaluate limit(2*x-4,x = 3)  

   MATRIX([[`Algebraic Manipulation`, Remarks], [________________________, ________________________], [limit(2*x-4,x = 3), `original limit`], [`` = limit(2*x,x = 3)-limit(4,x = 3), `applied difference rul...  

That is, limit(2*x-4,x = 3) = 2 .

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In addition to following the argument provided here you are encouraged to repeat these examples in the Limit  maplet [ Maplet Viewer] [ MapleNet]. To specify a problem in the Limit  maplet 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 Limit maplet. 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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Example 2: Use the Limit Laws to evaluate limit(x^2-3*x+6,x = 1)  

   MATRIX([[`Algebraic Manipulation`, Remarks], [________________________, ________________________], [limit(x^2-3*x+6,x = 1), `original limit`], [`` = limit(x^2,x = 1)-limit(3*x,x = 1)+limit(6,x = 1), `a...  

That is, limit(x^2-3*x+6,x = 1) = 4 .

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The final example involves the quotient rule. Note that the Limit  maplet applies the selected rule without checking any hypotheses. In this case the quotient rule is applied without verifying the denominator is not zero.

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Example 3: Use the Limit Laws to evaluate limit((t^2+5*t)*(t^10-6*t^5+2*t)/(t^4+4*t^2+4),t = -2)  

This problem is going to use the product and quotient rules. Following the steps in Examples 1 and 2, it is easily seen that:

  limit(t^2+5*t,t = -2) = -6  

  limit(t^10-6*t^5+2*t,t = -2) = 1212  

  limit(t^4+4*t^2+4,t = -2) = 36  

Because the first two limits exist , the Product Law can be applied to obtain

  limit((t^2+5*t)*(t^10-6*t^5+2*t),t = -2)  = -6*1212  

Now, because this limit exists and because

  limit(t^4+4*t^2+4,t = -2)  = 36 <> 0 ,

the Quotient Law can be applied. The result is that

  limit((t^2+5*t)*(t^10-6*t^5+2*t)/(t^4+4*t^2+4),t = -2)  = (-6)*1212/36  = -202.

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The Squeeze Theorem

The limit Limit(x^2*sin(1/x),x = 0)  involves a product, but the Product Law cannot be applied because Limit(sin(1/x),x = 0)  does not exist (see Example 7 in the Conceptual Understanding of the Limit lesson). This is particularly frustrating because it would be nice if we could argue that Limit(x^2,x = 0) = 0  and  0*(anything)=0. But, this is true only if we know something more about `anything`.

The Squeeze Theorem is a Limit Law that can be used is this type of situation.

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Theorem: The Squeeze Theorem

Let f, g, and h be functions satisfying:

  f(x)  <= g(x)  <= h(x) , for all x  near a  (except possibly at x = a ).

If Limit(f(x),x = a) = L  and Limit(h(x),x = a) = L , then Limit(g(x),x = a) = L .

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The idea behind the Squeeze Theorem is this: if you know a function's values are bounded above and below by functions with known limits at a point --- and these limits have the same value --- then the unknown limit exists and has the same value as the other limits.

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Example 4: Limit(x^2*sin(1/x),x = 0)  

As noted at the beginning of this lesson, because Limit(sin(1/x),x = 0)  does not exist (see Example 7 in the Conceptual Understanding of the Limit lesson), this limit cannot be evaluated using the Product Law.

A plot suggests what is happening for this limit:

>    g := x^2*sin(1/x):
plot( g, x=-1/2..1/2, color=blue, title="Plot of y=x^2*sin(1/x) on [-1/2,1/2]" );

[Maple Plot]

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The oscillations in the sine function are the reason Limit(sin(1/x),x = 0)   does not exist. Do the oscillations in the above graph mean Limit(x^2*sin(1/x),x = 0)   does not exist?

The fact that all values of the sine function are between -1 and 1 (inclusive) means that

-1 <= sin(1/x)  <= 1 for all x <> 0 .

>    plot( [-1,sin(1/x),1], x=-1/2..1/2, color=[red,blue,red], title="Plot showing that -1 <= sin(1/x) <= 1 near 0" );

[Maple Plot]

 Multiplying these inequalities by x^2  (a positive quantity for x <> 0 ) yields

  -x^2  <= x^2*sin(1/x)  <= x^2  for all x <> 0 .

>    f := -x^2:
h := x^2:
plot( [f,g,h], x=-1/2..1/2, color=[red,blue,red], title="Plot showing that -x^2 <= x^2*sin(1/x) <= x^2 near 0" );

[Maple Plot]

Now, because Limit(-x^2,x = 0) = 0  and Limit(x^2,x = 0) = 0  --- note that both limits exist and have the same limit --- the Squeeze Theorem allows us to conclude that

  Limit(x^2*sin(1/x),x = 0) = 0 .

Maple can confirm this result by direct computation:

>    q1 := Limit( x^2*sin(1/x), x=0 ):
q1 = value( q1 );

Limit(x^2*sin(1/x),x = 0) = 0

Additional examples involving the Squeeze Theorem will appear in the Limits of Trigonometric Functions lesson.

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Lesson Summary

The Limit Laws, including the Squeeze Theorem, provide rigorous tools for evaluating limits without  having to provide an epsilon-delta  proof for each and every limit that we encounter. The most important skill to develop during this lesson is the ability to apply the limit laws in an appropriate order to evaluate a limit. The Limit  maplet [ Maplet Viewer] [ MapleNet] provides an excellent tool to learn the names of the limit laws and how they can be applied to evaluate a limit.

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What's Next?

The ability to apply the Limit Laws is fundamental to success in calculus. Both differentiation ( Unit 2) and integration ( Unit 4) have analogues of the Limit Laws. For these reasons you are strongly encouraged to make extensive use of the online practice sessions prior to completing the online homework assignment and the textbook homework assignment.

The next lesson, Limits of Trigonometric Functions, will be our first extended application of the Limit Laws.

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