1.1.7

 


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1. An Example

 

Example 1.1

 

Find this limit:

 

 

Note

 

 

Fig. 1.1

 

As x approaches infinity, (10x + 1)/(2x 3) appears to approach 5.

 

The limit appears to be 5. Let's now find it formally.

 

Solution

EOS

 

Indeed the limit is 5.

 


C
onsider the following four trivial limits:

 

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3. Evaluation Techniques


Dominating Terms

Example 3.1

 

Determine:

 

 

Solution 1

EOS

 

Solution 2

EOS

In general:

 



 

Of course solution 2 is simpler and thus preferable to solution 1.

 

 

Example 3.2

 

Evaluate:

 

 

Solution

EOS

 

Example 3.3

 

Find:

 

 

Solution

EOS

 

Highest Power Of x In The Denominator

Let's find the same limit as in example 3.1 using a different technique.

 

Example 3.4

 

Find:

 

 

Solution

EOS


Thus, instead of using the quotient of the dominating terms, we can divide both the numerator and the denominator by
the highest power of
x in the denominator. The highest power of x in the denominator in this example is x3.

We note that the dominating-terms technique is simpler and faster than the highest-power-of-
x-in-the-denominator
technique. For fractions with long numerator and/or denominator, the dominating-terms technique is
preferable.

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Problems & Solutions


1. Find:




Solution



Note


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2. Determine:




Solution



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3. Evaluate:

 

 

if it exists.

 

Solution

 

 

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4. Find:

 

 

Solution

 

 

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5. Determine:




Solution



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