1.1.7

### 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.

###### EOS

Indeed the limit is 5.

C
onsider the following four trivial limits:

 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.

#### Problems & Solutions

1.  Find:

Solution

Note

2.  Determine:

Solution

3.  Evaluate:

if it exists.

4.  Find:

5.  Determine:

Solution