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work:

|x^2 + 3 - 7| < .05
|x^2 - 4| < .05
-.05 < x^2 - 4 < .05
3.95 < x^2 < 4.05
1.98746... < x < 2.01246...
-.01253... < x - 2 < .01246






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[ see whether it is true that, whenever |x-2| < 0.01253, we
will have |x^2-4| < 0.05.]
For a quick test, we can just try making |x-2| = 0.01252 and see if it works. In one direction, we have x = 2.01252, x^2 = 4.0502367504, and x^2-4 = 0.0502367504. This is NOT less than 0.05, so we see that it doesn't work.

Now, how could we have thought through this to see why we should
choose delta as 0.01246? Look back at what you found:

   -.01253 < x - 2 < .01246

You showed that this is equivalent to |x^2 - 4| < 0.05 (given that x > 0). You want to choose a delta so that this will be true whenever |x-2| < delta. Notice that the latter is the same as

   -delta < x - 2 < delta

If you pick delta = 0.01246, then this interval is inside the interval from -0.01253<x-2<0.01246, so that any x within delta of 2 does what you want. If you choose delta = 0.01253, then

   -0.01253 < x - 2 < 0.01253

does NOT imply

   -0.01253 < x - 2 < 0.01246

because the former includes some numbers that are not in the latter.

Do you see how the reasoning goes? Essentially we want a symmetrical interval that is INSIDE the interval equivalent to the epsilon goal, so that x being within delta IMPLIES that f(x) is within epsilon. So we can choose any delta LESS THAN OR EQUAL TO 0.01246.


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http://www.geocities.com/mathdepot/definit.htm


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