Let $g\left(t\right)=\frac{t-9}{\sqrt{t}-3}$.
Make two tables, one showing values of $g$ for $t=8.9,8.99$, and $8.999$ and one showing values of $g$ for $t=9.1,9.01$, and $9.001$.

Verified step by step guidance

1. Identify the function g(t) = \frac{t-9}{\sqrt{t}-3}. This function is undefined at t = 9 because both the numerator and the denominator become zero, leading to an indeterminate form.
2. To understand the behavior of g(t) as t approaches 9, we will evaluate the function for values of t slightly less than 9 and slightly greater than 9.
3. Create the first table for values of t less than 9: Calculate g(t) for t = 8.9, 8.99, and 8.999. For each value, substitute t into the function and simplify to find g(t).
4. Create the second table for values of t greater than 9: Calculate g(t) for t = 9.1, 9.01, and 9.001. Again, substitute each value of t into the function and simplify to find g(t).
5. Analyze the results from both tables to observe the trend of g(t) as t approaches 9 from both sides. This will help in understanding the limit of g(t) as t approaches 9.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this question, evaluating the function g(t) as t approaches 9 is crucial, as it may involve determining the limit to handle any indeterminate forms that arise.

Rational Functions

A rational function is a ratio of two polynomials. The function g(t) = (t - 9) / (√t - 3) is a rational function, and understanding its behavior, especially around points where the denominator may approach zero, is essential for analyzing its values and limits.

Table of Values

Creating a table of values involves calculating the output of a function for specific input values. In this question, constructing tables for g(t) at values close to 9 helps visualize the function's behavior and aids in understanding its limit and continuity around that point.