Make an input-output table in order to investigate the behavior of ( f(x) = \frac{\sqrt{x} - 3}{x - 9} ) as ( x ) approaches 9 from the left and the right.

We are tasked with investigating the behavior of the function \( f(x) = \frac{\sqrt{x} - 3}{x - 9} \) as \( x \) approaches 9 from both the left and the right. To do this, we'll create an input-output table by plugging in values of \( x \) that approach 9 from both sides. This will help us understand how \( f(x) \) behaves near \( x = 9 \).

### Step 1: Function Analysis
The function \( f(x) \) seems problematic at \( x = 9 \) because it results in a \( \frac{0}{0} \) indeterminate form. To explore the behavior, we will examine values of \( x \) that are slightly less than 9 (approaching from the left) and slightly greater than 9 (approaching from the right).

### Step 2: Input-Output Table
We will compute \( f(x) \) for values of \( x \) near 9:

#### Approaching 9 from the Left (\( x < 9 \)):
| \( x \)        | \( f(x) \)                           |
|----------------|---------------------------------------|
| 8.9            | \( \frac{\sqrt{8.9} - 3}{8.9 - 9} \) |
| 8.99           | \( \frac{\sqrt{8.99} - 3}{8.99 - 9} \)|
| 8.999          | \( \frac{\sqrt{8.999} - 3}{8.999 - 9}\)|
| 8.9999         | \( \frac{\sqrt{8.9999} - 3}{8.9999 - 9}\) |

#### Approaching 9 from the Right (\( x > 9 \)):
| \( x \)        | \( f(x) \)                           |
|----------------|---------------------------------------|
| 9.1            | \( \frac{\sqrt{9.1} - 3}{9.1 - 9} \) |
| 9.01           | \( \frac{\sqrt{9.01} - 3}{9.01 - 9} \)|
| 9.001          | \( \frac{\sqrt{9.001} - 3}{9.001 - 9}\)|
| 9.0001         | \( \frac{\sqrt{9.0001} - 3}{9.0001 - 9}\) |

Now let's calculate the values of \( f(x) \) numerically for these inputs. I will compute these values next.Here are the computed values for \( f(x) = \frac{\sqrt{x} - 3}{x - 9} \) as \( x \) approaches 9 from both the left and the right:

#### Approaching 9 from the Left (\( x < 9 \)):
| \( x \)        | \( f(x) \)              |
|----------------|-------------------------|
| 8.9            | 0.16713                 |
| 8.99           | 0.16671                 |
| 8.999          | 0.16667                 |
| 8.9999         | 0.16667                 |

#### Approaching 9 from the Right (\( x > 9 \)):
| \( x \)        | \( f(x) \)              |
|----------------|-------------------------|
| 9.1            | 0.16621                 |
| 9.01           | 0.16662                 |
| 9.001          | 0.16666                 |
| 9.0001         | 0.16667                 |

### Observation:
As \( x \) approaches 9 from both the left and the right, the values of \( f(x) \) approach approximately 0.16667. This suggests that the limit of \( f(x) \) as \( x \to 9 \) is approximately 0.16667.

Would you like a more detailed explanation of this behavior, or do you have any other questions?

### 5 Related Questions:
1. What is the formal limit of \( f(x) \) as \( x \to 9 \)?
2. How do we handle indeterminate forms like \( \frac{0}{0} \)?
3. What techniques can be used to simplify such expressions?
4. How can you verify if a function has a removable discontinuity?
5. What happens to \( f(x) \) for values much farther from 9?

### Tip:
In problems involving limits and indeterminate forms, factoring or using the conjugate can often help simplify the expression for easier evaluation.

Explore the behavior of the function f(x) = (√x - 3) / (x - 9) as x approaches 9 from the left and the right. Learn how to handle indeterminate forms and understand the limits of functions with an input-output table approach. Suitable for Grades 11-12.

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