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.