https://youtu.be/7xDC8aV-mMg

In this video I continue with the wonderful world of limits and go over a more difficult example using the precise definition of a limit. In this case I show how to prove the limit of x<sup>2</sup> as x approaches 3 is equal to 9 using a pretty clever method. For more complicated functions, using the precise definition to prove limits becomes increasingly more difficult. But luckily we can simply prove them using the limit laws which I went over in my earlier videos (see video links below). But those limit laws need to be proven and I will prove each one in my videos to come.

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# Precise Definition of a Limit – Example 3

![Precise Definition of a Limit  Example 3.jpg](https://files.peakd.com/file/peakd-hive/mes/QoEiJ9Iv-Precise20Definition20of20a20Limit20-20Example203.jpg)
	
## Precise Definition

Let f be a function defined on some open interval that contains the number ‘a’, except possibly at ‘a’ itself. 

Then we can say that the **limit of f(x) as x approaches ‘a’ is ‘L’**, and we write:

![image.png](https://files.peakd.com/file/peakd-hive/mes/w2dTD55h-image.png)

If for every number ε > 0 there is a number δ > 0 such that:

![image.png](https://files.peakd.com/file/peakd-hive/mes/bZC2uz4x-image.png)

## Example

![image.png](https://files.peakd.com/file/peakd-hive/mes/wor7c4oC-image.png)

### Solution

![image.png](https://files.peakd.com/file/peakd-hive/mes/YupkEwVL-image.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/s7dhzXtj-image.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/sUdSPUdB-image.png)

## Important Notes From This Example

- Not always easy to prove the limit using the precise definition of a limit
  - In fact, complicated functions like **f(x) = (6x<sup>2</sup> – 8x + 9)/(2x<sup>2</sup> – 1)** require a great deal of ingenuity
- Fortunately, we can actually prove limits such as these using the Limit Laws which I covered earlier
   - All we need to do first is prove each limit law using the precise definition of a limit (in my later videos)