The point $$ P(3,-3) $$ lies on the curve $$ y=\frac{3}{2-x} $$. (a) If $$ Q $$ is the point $$ \left(x, \frac{3}{2-x}\right) $$, find the slope of the secant line $$ P Q $$ (correct to six decimal places) for the following values of $$ x $$. (I) 2.9 $$ m_{P Q}= $$ $$ \qquad $$ (ii) 2.99 $$ m_{P Q}=1 \quad x $$ (iii) 2.999 $$ m_{P Q}= $$ $$ \qquad $$ (iv) 2.9999 $$ m_{P Q}= $$ $$ \qquad $$ (v) 3.1 $$ m_{P Q}= $$ $$ \qquad $$ (vi) 3.01 $$ m_{P O}= $$ $$ \qquad $$ (vii) 3.001 $$ m_{P Q}= $$ $$ \qquad $$ (viii) 3.0001 $$ m_{P Q}= $$ $$ \qquad $$ (b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at $$ P(3,-3) $$. $$ m= $$ $$ \qquad $$ (c) Using the slope from part (b), find an equation of the tangent line to the curve at $$ P(3,-3) $$. $$ \square $$

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To solve the problem, we will follow the steps outlined in the prompt.

### Part (a)

We need to find the slope of the secant line $$ PQ $$ for various values of $$ x $$. The slope $$ m_{PQ} $$ of the secant line between points $$ P(3, -3) $$ and $$ Q\left(x, \frac{3}{2-x}\right) $$ is given by the formula:

$$
m_{PQ} = \frac{y_Q - y_P}{x_Q - x_P} = \frac{\frac{3}{2-x} - (-3)}{x - 3}
$$

This simplifies to:

$$
m_{PQ} = \frac{\frac{3}{2-x} + 3}{x - 3}
$$

Now, we will calculate $$ m_{PQ} $$ for the specified values of $$ x $$.

1. **For $$ x = 2.9 $$**:
   $$
   m_{PQ} = \frac{\frac{3}{2-2.9} + 3}{2.9 - 3}
   $$
   $$
   = \frac{\frac{3}{-0.9} + 3}{-0.1} = \frac{-3.3333 + 3}{-0.1} = \frac{-0.3333}{-0.1} = 3.3333
   $$

2. **For $$ x = 2.99 $$**:
   $$
   m_{PQ} = \frac{\frac{3}{2-2.99} + 3}{2.99 - 3}
   $$
   $$
   = \frac{\frac{3}{-0.99} + 3}{-0.01} = \frac{-3.0303 + 3}{-0.01} = \frac{-0.0303}{-0.01} = 3.0303
   $$

3. **For $$ x = 2.999 $$**:
   $$
   m_{PQ} = \frac{\frac{3}{2-2.999} + 3}{2.999 - 3}
   $$
   $$
   = \frac{\frac{3}{-0.999} + 3}{-0.001} = \frac{-3.003003 + 3}{-0.001} = \frac{-0.003003}{-0.001} = 3.003003
   $$

4. **For $$ x = 2.9999 $$**:
   $$
   m_{PQ} = \frac{\frac{3}{2-2.9999} + 3}{2.9999 - 3}
   $$
   $$
   = \frac{\frac{3}{-0.9999} + 3}{-0.0001} = \frac{-3.00030003 + 3}{-0.0001} = \frac{-0.00030003}{-0.0001} = 3.0003003
   $$

5. **For $$ x = 3.1 $$**:
   $$
   m_{PQ} = \frac{\frac{3}{2-3.1} + 3}{3.1 - 3}
   $$
   $$
   = \frac{\frac{3}{-1.1} + 3}{0.1} = \frac{-2.72727 + 3}{0.1} = \frac{0.27273}{0.1} = 2.7273
   $$

6. **For $$ x = 3.01 $$**:
   $$
   m_{PQ} = \frac{\frac{3}{2-3.01} + 3}{3.01 - 3}
   $$
   $$
   = \frac{\frac{3}{-1.01} + 3}{0.01} = \frac{-2.9703 + 3}{0.01} = \frac{0.0297}{0.01} = 2.97
   $$

7. **For $$ x = 3.001 $$**:
   $$
   m_{PQ} = \frac{\frac{3}{2-3.001} + 3}{3.001 - 3}
   $$
   $$
   = \frac{\frac{3}{-1.001} + 3}{0.001} = \frac{-2.997002 + 3}{0.001} = \frac{0.002998}{0.001} = 2.998
   $$

8. **For $$ x = 3.0001 $$**:
   $$
   m_{PQ} = \frac{\frac{3}{2-3.0001} + 3}{3.0001 - 3}
   $$
   $$
   = \frac{\frac{3}{-1.0001} + 3}{0.0001} = \frac{-2.99970003 + 3}{0.0001} = \frac{0.00029997}{0.0001} = 2.9997
   $$

### Summary of Results for Part (a)

- $$ m_{PQ} $$ for $$ x = 2.9 $$: $$ 3.333333 $$
- $$ m_{PQ} $$ for $$ x = 2.99 $$: $$ 3.030303 $$
- $$ m_{PQ} $$ for $$ x = 2.999 $$: $$ 3.003003 $$
- $$ m_{PQ} $$ for $$ x = 2.9999 $$: $$ 3.000300 $$
- $$ m_{PQ} $$ for $$ x = 3.1 $$: $$ 2.727273 $$
- $$ m_{PQ} $$ for $$ x = 3.01 $$: $$ 2.970000 $$
- $$ m_{PQ} $$ for $$ x = 3.001 $$: $$ 2.998001 $$
- $$ m_{PQ} $$ for $$ x = 3.0001 $$: $$ 2.999700 $$

### Part (b)

From the results of part (a), as $$ x $$ approaches $$ 3 $$, the slope $$ m_{PQ} $$ approaches $$ 3 $$. Therefore, we can guess that the slope of the tangent line at point $$ P(3, -3) $$ is:

$$
m = 3
$$

### Part (c)

To find the equation of the tangent line at point $$ P(3, -3) $$ with slope $$ m = 3 $$, we can use the point-slope form of the equation of a line:

$$
y - y_1 = m(x - x_1)
$$

Substituting $$ m = 
**Part (a):** 1. **For $$ x = 2.9 $$:** $$ m_{PQ} = 3.333333 $$ 2. **For $$ x = 2.99 $$:** $$ m_{PQ} = 3.030303 $$ 3. **For $$ x = 2.999 $$:** $$ m_{PQ} = 3.003003 $$ 4. **For $$ x = 2.9999 $$:** $$ m_{PQ} = 3.000300 $$ 5. **For $$ x = 3.1 $$:** $$ m_{PQ} = 2.727273 $$ 6. **For $$ x = 3.01 $$:** $$ m_{PQ} = 2.970000 $$ 7. **For $$ x = 3.001 $$:** $$ m_{PQ} = 2.998001 $$ 8. **For $$ x = 3.0001 $$:** $$ m_{PQ} = 2.999700 $$ **Part (b):** The slope of the tangent line at $$ P(3, -3) $$ is **3**. **Part (c):** The equation of the tangent line at $$ P(3, -3) $$ with slope $$ m = 3 $$ is: $$ y - (-3) = 3(x - 3) $$ $$ y + 3 = 3x - 9 $$ $$ y = 3x - 12 $$

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