Epsilon-delta definition of limit

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If we make a statement like , what we mean is that as gets closer and closer to 5, the function gets closer and closer to 10. You might ask the question (if this were a less obvious example) "How do you know that gets closer and closer to 10 as gets closer and closer to 5? If that really is true, show me values centered around 5 for which is within ½ of 10." (see picture below, not to scale)

so eventually we should get to the
point where is within ½ of 10

 is getting closer and closer to 5

I take a look at your statement " is within ½ of 10" and realize that this ...