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^{2} 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

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:

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

## Example

### Solution

## 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**require a great deal of ingenuity^{2}– 8x + 9)/(2x^{2}– 1)

- In fact, complicated functions like
- 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)