Precise Definition of a Limit - Example 3

in MES Science2 months ago

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 x2 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

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:

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

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Example

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Solution

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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) = (6x2 – 8x + 9)/(2x2 – 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)
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