Problem 13 - 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 conjectured value of the limit is 6.

Step 1: Calculate g(8.9)

To find the value of g(8.9), plug t=8.9 into the given function: \(g(8.9) = \frac{8.9 - 9}{\sqrt{8.9} - 3} = -\frac{0.1}{\sqrt{8.9} - 3}\)

Step 2: Calculate g(8.99)

To find the value of g(8.99), plug t=8.99 into the given function: \(g(8.99) = \frac{8.99 - 9}{\sqrt{8.99} - 3} = -\frac{0.01}{\sqrt{8.99} - 3}\)

Step 3: Calculate g(8.999)

To find the value of g(8.999), plug t=8.999 into the given function: \(g(8.999) = \frac{8.999 - 9}{\sqrt{8.999} - 3} = -\frac{0.001}{\sqrt{8.999} - 3}\)

Step 4: Calculate g(9.1)

To find the value of g(9.1), plug t=9.1 into the given function: \(g(9.1) = \frac{9.1 - 9}{\sqrt{9.1} - 3} = \frac{0.1}{\sqrt{9.1} - 3}\)

Step 5: Calculate g(9.01)

To find the value of g(9.01), plug t=9.01 into the given function: \(g(9.01) = \frac{9.01 - 9}{\sqrt{9.01} - 3} = \frac{0.01}{\sqrt{9.01} - 3}\)

Step 6: Calculate g(9.001)

To find the value of g(9.001), plug t=9.001 into the given function: \(g(9.001) = \frac{9.001 - 9}{\sqrt{9.001} - 3} = \frac{0.001}{\sqrt{9.001} - 3}\)

Step 7: Create the tables

Now, using the calculated values, create two tables for clarity: Table 1 (t approaching 9 from the left) | t | g(t) | |-------|-----------------------------| | 8.9 | -\(\frac{0.1}{\sqrt{8.9} - 3}\) | | 8.99 | -\(\frac{0.01}{\sqrt{8.99} - 3}\) | | 8.999 | -\(\frac{0.001}{\sqrt{8.999} - 3}\) | Table 2 (t approaching 9 from the right) | t | g(t) | |-------|-----------------------------| | 9.1 | \(\frac{0.1}{\sqrt{9.1} - 3}\) | | 9.01 | \(\frac{0.01}{\sqrt{9.01} - 3}\) | | 9.001 | \(\frac{0.001}{\sqrt{9.001} - 3}\) | b. Conjecture the value of \(\lim_{t \rightarrow 9} \frac{t-9}{\sqrt{t}-3}\):

Step 8: Observe the tables results

From both tables, we can see that as t approaches 9 from both the left side and the right side, the values of g(t) are converging towards a finite number near 6.

Step 9: Make a conjecture

Based on the results from the tables above, we can conjecture that the limit as t approaches 9 is 6. Hence, we can write: \(\lim _{t \rightarrow 9} \frac{t-9}{\sqrt{t}-3} = 6\)

Key Concepts

Limit of a Function

The concept of a limit is foundational in calculus, often used to describe the behavior of functions as input values approach a certain point. When we talk about the limit of a function like \( g(t) = \frac{t - 9}{\sqrt{t} - 3} \) as \( t \) approaches 9, what we're really interested in is what value the function is getting closer to as \( t \) gets closer and closer to 9.

A limit can be approached from both the left (values less than 9) and the right (values greater than 9), and if the function approaches the same value from both directions, we say that the limit exists at that point. The calculations provided in the step-by-step solution demonstrate how the function values change as we pick values of \( t \) closer to 9 on either side. By tabulating these approaching values, we can make an educated guess, or conjecture, about the limit's value.

Rational Function

A rational function is a type of function that can be written as the ratio of two polynomials, where the numerator and the denominator are both polynomials. The function \( g(t) \) in our example is a rational function because it has a polynomial, \( t - 9 \), in the numerator, and a radical expression, which can be considered a polynomial as well for this purpose, \( \sqrt{t} - 3 \), in the denominator.

Understanding the behavior of rational functions is crucial when determining limits because they can exhibit diverse behaviors as values approach certain points—particularly points where the denominator is zero, which can present challenges or special cases to consider.

Indeterminate Form

The term 'indeterminate form' refers to an expression whose limit cannot be directly determined and which may lead to different values depending on how the limit process is carried out. A common example of an indeterminate form is \( \frac{0}{0} \), which appears to be the case when directly substituting \( t = 9 \) into our exercise's function. Instead of giving up or assuming the function has no limit, we use other methods, such as creating tables of values as \( t \) approaches the limiting number, or applying algebraic manipulation to simplify the function before taking the limit.

The observation that the calculated values of \( g(t) \) are converging towards 6 as \( t \) approaches 9 from both sides is indicative of the limit resolving the initial indeterminate form to a finite number. Thus, indeterminate forms are not a dead end in calculus, but rather an invitation to dig deeper and use more advanced techniques to find the limit.