# How do you prove that the limit of #(x+2)/(x-3) = -1/4# as x approaches -1 using the epsilon delta proof?

See the section below.

Preliminary analysis

By definition,

Look at the thing we want to make small. Rewrite this, looking for the thing we control.

Now we need to actually write up the proof:

Proof

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To prove that the limit of (x+2)/(x-3) is -1/4 as x approaches -1 using the epsilon-delta proof, we need to show that for any given epsilon greater than 0, there exists a delta greater than 0 such that whenever 0 < |x - (-1)| < delta, then |(x+2)/(x-3) - (-1/4)| < epsilon.

Let's proceed with the proof:

Given epsilon > 0, we need to find a suitable delta > 0.

First, let's simplify the expression |(x+2)/(x-3) - (-1/4)|:

|(x+2)/(x-3) + 1/4| = |(4(x+2) + (x-3))/(4(x-3))| = |(5x + 5)/(4x - 12)|

Now, we can proceed with the proof:

Choose delta = epsilon/20.

Assume that 0 < |x - (-1)| < delta.

Then, we have:

|x + 1| < delta

|x + 1| < epsilon/20

|x + 1| * |4x - 12| < (epsilon/20) * |4x - 12|

|(x + 1)(4x - 12)| < (epsilon/20) * |4x - 12|

|4x^2 - 8x - 12x + 24| < (epsilon/20) * |4x - 12|

|4x^2 - 20x + 24| < (epsilon/20) * |4x - 12|

4|x^2 - 5x + 6| < (epsilon/20) * |4x - 12|

4|x - 2||x - 3| < (epsilon/20) * |4x - 12|

Since we have assumed that 0 < |x - (-1)| < delta, we can also assume that |x - 2| < 3 and |x - 3| < 3.

Therefore, we have:

4|x - 2||x - 3| < 12|x - 2|

12|x - 2| < (epsilon/20) * |4x - 12|

12|x - 2| < (epsilon/20) * 4|x - 3|

3|x - 2| < (epsilon/5) * |x - 3|

Now, we can choose a suitable value for delta. Let's choose delta = min(1, epsilon/5).

If we assume that 0 < |x - (-1)| < delta, then we have:

|x - 2| < delta

|x - 2| < min(1, epsilon/5)

3|x - 2| < 3 * min(1, epsilon/5)

3|x - 2| < min(3, 3 * epsilon/5)

3|x - 2| < min(3, epsilon)

Therefore, we can conclude that for any given epsilon > 0, if we choose delta = min(1, epsilon/5), then whenever 0 < |x - (-1)| < delta, we have:

3|x - 2| < min(3, epsilon)

And since 3|x - 2| < min(3, epsilon), we can also conclude that:

|(x+2)/(x-3) - (-1/4)| < epsilon

Hence, the limit of (x+2)/(x-3) as x approaches -1 is -1/4, as proven using the epsilon-delta proof.

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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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