# How do you prove the statement lim as x approaches 2 for # (x^2 - 3x) = -2# using the epsilon and delta definition?

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Initial Analysis

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Proof

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To prove the statement lim as x approaches 2 for (x^2 - 3x) = -2 using the epsilon and delta definition, we need to show that for any given epsilon greater than 0, there exists a delta greater than 0 such that if 0 < |x - 2| < delta, then |(x^2 - 3x) - (-2)| < epsilon.

Let's proceed with the proof:

Given the function f(x) = x^2 - 3x, we want to find a delta such that if 0 < |x - 2| < delta, then |(x^2 - 3x) - (-2)| < epsilon.

First, let's simplify the expression |(x^2 - 3x) - (-2)|: |(x^2 - 3x) + 2| = |x^2 - 3x + 2|

Now, we can factorize the expression inside the absolute value: |x^2 - 3x + 2| = |(x - 2)(x - 1)|

Since we are interested in the behavior of the function as x approaches 2, we can assume that delta is small enough such that 0 < |x - 2| < delta implies 0 < |x - 2| < 1. This allows us to make the following observations:

- If 0 < x < 3, then 0 < x - 1 < 2.
- If 1 < x < 3, then 0 < x - 2 < 1.

Using these observations, we can rewrite the expression |(x - 2)(x - 1)| as follows:

- If 0 < x < 3, then |(x - 2)(x - 1)| = |x - 2||x - 1|.
- If 1 < x < 3, then |(x - 2)(x - 1)| = |x - 2||x - 1| = -(x - 2)(x - 1).

Now, let's consider the case when 0 < x < 3:

|x - 2||x - 1| = |x - 2||x - 1| < 2|x - 2|

We can choose delta = min(1, epsilon/2). Then, if 0 < |x - 2| < delta, we have:

|x - 2||x - 1| < 2|x - 2| < 2(delta) = 2(min(1, epsilon/2)) = epsilon

Therefore, we have shown that for any given epsilon > 0, there exists a delta > 0 such that if 0 < |x - 2| < delta, then |(x^2 - 3x) - (-2)| < epsilon. This proves the statement lim as x approaches 2 for (x^2 - 3x) = -2 using the epsilon and delta definition.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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