Problem 9 - Calculus: Early Transcendentals

Question

Let \(g(t)=\frac{t-9}{\sqrt{t}-3}\) a. 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. b. Make a conjecture about the value of \(\lim _{t \rightarrow 9} \frac{t-9}{\sqrt{t}-3}\).

Step-by-Step Solution

Answer

Answer: The calculated values of g(t) suggest that the limit of the function g(t) = (t-9)/(√t - 3) as t approaches 9 might be 6.

Step 1: Create the first table for values near 9

To find the values of \(g(t)\) for \(t=8.9, 8.99, 8.999\), simply plug in each value of \(t\) into the function and calculate the corresponding values of \(g(t)\). 1. For \(t=8.9\), \(g(8.9)=\frac{8.9-9}{\sqrt{8.9}-3}\approx -6.11\) 2. For \(t=8.99\), \(g(8.99)=\frac{8.99-9}{\sqrt{8.99}-3}\approx -6.01\) 3. For \(t=8.999\), \(g(8.999)=\frac{8.999-9}{\sqrt{8.999}-3}\approx -6.00\) The first table is: | t | g(t) | |-------|-------| | 8.9 | -6.11 | | 8.99 | -6.01 | | 8.999 | -6.00 |

Step 2: Create the second table for values near 9

Now, we will find the values of \(g(t)\) for \(t=9.1, 9.01, 9.001\) by plugging in each value of \(t\) into the function and calculating the corresponding values of \(g(t)\). 1. For \(t=9.1\), \(g(9.1)=\frac{9.1-9}{\sqrt{9.1}-3}\approx 6.22\) 2. For \(t=9.01\), \(g(9.01)=\frac{9.01-9}{\sqrt{9.01}-3}\approx 6.03\) 3. For \(t=9.001\), \(g(9.001)=\frac{9.001-9}{\sqrt{9.001}-3}\approx 6.00\) The second table is: | t | g(t) | |-------|-------| | 9.1 | 6.22 | | 9.01 | 6.03 | | 9.001 | 6.00 |

Step 3: Make a conjecture about the limit

By observing the values in both tables, we can see that as \(t\) approaches 9, the values of \(g(t)\) are getting closer to 6. This suggests that the limit of the function as \(t\) approaches 9 could be 6. Formally, we can conjecture that: $$ \lim_{t \rightarrow 9} \frac{t-9}{\sqrt{t}-3} = 6 $$

Key Concepts

Understanding Function Evaluation

To solve problems like evaluating the limit of a function, the first step is often to substitute various values into the function. This process is known as "function evaluation." In the given exercise, we are asked to evaluate the function \(g(t) = \frac{t-9}{\sqrt{t}-3}\) for values near \(t=9\). By substituting different values for \(t\), you can observe how the function behaves as \(t\) gets very close to 9. Here's a quick guide on function evaluation:

• Substitute a chosen value into the function in place of the variable \(t\).
• Perform the arithmetic operation according to the function definition.
• Record the result, which gives you \(g(t)\) for that specific \(t\).

This method helps to estimate a pattern or a limit, forming the foundation for generating tables of values and making conjectures on limits.

Creating Tables of Values

When dealing with limits, making "tables of values" is a useful technique to see how a function behaves as its input approaches a particular point. In this specific exercise, tables of values were created to observe how the function \(g(t)\) changes as \(t\) gets closer to 9 from both directions (less than and greater than 9).Here’s how you can create your own table of values:

• Select values that are close to the desired limit point (e.g., before and after \(t=9\)).
• Use function evaluation to calculate \(g(t)\) for each selected value.
• Record these values in a formatted table to easily compare them.

In this case, tables were filled with results from values just below and just above 9. This method lets us visualize how the outputs approach a specific value as \(t\) nears 9, making it easier to predict potential limits.

Formulating a Conjecture About Limits

Once you have your tables of values, the next step is "making a conjecture about the limits." In the context of the given problem, you look at how the values of \(g(t)\) change as \(t\) gets closer to the target point, 9. From the exercise's tables, the function values near \(t=9\) converge towards 6.Here's how you can formulate your own conjecture:

• Look for a pattern or tendency in your table of values.
• Consider the behavior of these values from both sides of the limit point (approaching from below and above \(t=9\)).
• Make an educated guess or conjecture regarding the limit based on these observations.

In this problem, by observing the symmetry and closeness of values from both sides, the conjecture \(\lim_{t \rightarrow 9} \frac{t-9}{\sqrt{t}-3} = 6\) was formed. This approach helps in understanding potential limits in different mathematical situations, supporting deeper insights into the function's behavior.