Math Problem Statement

Let g(t)equalsStartFraction t minus 49 Over StartRoot t EndRoot minus 7 EndFraction . a. Make two tables, one showing the values of g for tequals48.9, 48.99, and 48.999 and one showing values of g for tequals49.1, 49.01, and 49.001. b. Make a conjecture about the value of ModifyingBelow lim With t right arrow 49 StartFraction t minus 49 Over StartRoot t EndRoot minus 7 EndFraction . Question content area bottom Part 1 a. Make a table showing the values of g for tequals48.9, 48.99, and 48.999. t 48.9 48.99 48.999g(t) enter your response here enter your response here enter your response here(Round to four decimal places.) Use this example of a similiar problem to answer step by step: Let g(t)equalsStartFraction t minus 121 Over StartRoot t EndRoot minus 11 EndFraction . a. Make two tables, one showing the values of g for tequals120.9, 120.99, and 120.999 and one showing values of g for tequals121.1, 121.01, and 121.001. b. Make a conjecture about the value of ModifyingBelow lim With t right arrow 121 StartFraction t minus 121 Over StartRoot t EndRoot minus 11 EndFraction . Question content area bottom Part 1 a. Make a table showing the values of g for tequals120.9, 120.99, and 120.999 . Evaluate g left parenthesis 120.9 right parenthesis, rounding to four decimal places. g left parenthesis 120.9 right parenthesis equals StartFraction 120.9 minus 121 Over StartRoot 120.9 EndRoot minus 11 EndFraction equals 21.9955 Part 2 Evaluate g(120.99), rounding to four decimal places. g left parenthesis 120.99 right parenthesis equals StartFraction 120.99 minus 121 Over StartRoot 120.99 EndRoot minus 11 EndFraction equals 21.9995 Part 3 Evaluate g(120.999), rounding to four decimal places. g left parenthesis 120.999 right parenthesis equals StartFraction 120.999 minus 121 Over StartRoot 120.999 EndRoot minus 11 EndFraction equals 22.0000 Part 4Thus, the table of values of g for tequals120.9 comma 120.99, and 120.999 is given below. t 120.9 120.99 120.999g(t) 21.9955 21.9995 22.0000 Part 5 The values of g for tequals121.1, 121.01, and 121.001 are found similarly. Substitute these values into the function g(t)equalsStartFraction t minus 121 Over StartRoot t EndRoot minus 11 EndFraction and simplify, rounding to four decimal places. t 121.1 121.01 121.001g(t) StartFraction 121.1 minus 121 Over StartRoot 121.1 EndRoot minus 11 EndFraction equals22.0045 StartFraction 121.01 minus 121 Over StartRoot 121.01 EndRoot minus 11 EndFraction equals22.0005 StartFraction 121.001 minus 121 Over StartRoot 121.001 EndRoot minus 11 EndFraction equals22.0001 Part 6Thus, the two tables are as follows. t 120.9 120.99 120.999g(t) 21.9955 21.9995 22.0000 t 121.1 121.01 121.001g(t) 22.0045 22.0005 22.0001 Part 7 b. Make a conjecture about the value of ModifyingBelow lim With t right arrow 121 StartFraction t minus 121 Over StartRoot t EndRoot minus 11 EndFraction . Use the values just to the left and the right of tequals121 to approximate the limit at t equals 121. If the values do not approach a single value as t approaches 121, the limit does not exist. In this case, the values of g(t) both above and below tequals121 are approaching 22. Thus, ModifyingBelow lim With t right arrow 121 StartFraction t minus 121 Over StartRoot t EndRoot minus 11 EndFraction is likely equal to 22.

Solution

To solve the problem, let's break it down into steps based on the example given. We'll follow the same approach for the function:

$g(t) = \frac{t - 49}{\sqrt{t} - 7}$

Part 1: Make the tables

Table 1: Values of $g(t)$ for $t = 48.9, 48.99, 48.999$

We will evaluate the function $g(t)$ for these values of $t$ and round to four decimal places.

For $t = 48.9$ : $g(48.9) = \frac{48.9 - 49}{\sqrt{48.9} - 7} = \frac{-0.1}{\sqrt{48.9} - 7}$ Calculate the square root of $48.9$ and subtract 7 to get the denominator.
For $t = 48.99$ : $g(48.99) = \frac{48.99 - 49}{\sqrt{48.99} - 7}$
For $t = 48.999$ : $g(48.999) = \frac{48.999 - 49}{\sqrt{48.999} - 7}$

Table 2: Values of $g(t)$ for $t = 49.1, 49.01, 49.001$

Similarly, we'll evaluate the function for these values of $t$ .

For $t = 49.1$ : $g(49.1) = \frac{49.1 - 49}{\sqrt{49.1} - 7}$
For $t = 49.01$ : $g(49.01) = \frac{49.01 - 49}{\sqrt{49.01} - 7}$
For $t = 49.001$ : $g(49.001) = \frac{49.001 - 49}{\sqrt{49.001} - 7}$

We will now compute the values for these calculations.

Let me go ahead and perform these calculations. math

Function for g(t)

def g(t): return (t - 49) / (math.sqrt(t) - 7)

Values for the tables

t_values_1 = [48.9, 48.99, 48.999] t_values_2 = [49.1, 49.01, 49.001]

Calculate g(t) for each value

g_values_1 = [round(g(t), 4) for t in t_values_1] g_values_2 = [round(g(t), 4) for t in t_values_2]

g_values_1, g_values_2Here are the values of $g(t)$ for the tables:

Table 1: Values of $g(t)$ for $t = 48.9, 48.99, 48.999$

$t$	48.9	48.99	48.999
$g(t)$	13.9929	13.9993	13.9999

Table 2: Values of $g(t)$ for $t = 49.1, 49.01, 49.001$

$t$	49.1	49.01	49.001
$g(t)$	14.0071	14.0007	14.0001

Part 2: Conjecture about the limit

Looking at the values of $g(t)$ as $t$ approaches 49 from both sides:

From $t = 48.9, 48.99, 48.999$ , the function values approach $13.9999$ .
From $t = 49.1, 49.01, 49.001$ , the function values approach $14.0001$ .

Thus, the function values are getting closer to 14 as $t$ approaches 49, suggesting that the limit is approximately:

$\lim_{t \to 49} \frac{t - 49}{\sqrt{t} - 7} \approx 14$

Would you like further details or clarification on any part of this process? Here's a related set of questions:

How can you use the concept of limits to estimate values of a function as $t$ approaches a point?
What is the significance of the values approaching 14 in terms of continuity?
What happens if the limit from one side does not match the limit from the other side?
How would you estimate a limit for functions that don't behave smoothly around a certain point?
Can this method be applied to other rational functions involving square roots?

Tip: When computing limits for functions with square roots, always check if you can simplify the expression, for instance by multiplying by a conjugate if needed.

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Math Problem Analysis

Mathematical Concepts

Limits
Rational Functions
Square Roots
Conjectures

Formulas

g(t) = (t - 49) / (sqrt(t) - 7)

Theorems

Limit of a Rational Function
Conjecture for Limits

Suitable Grade Level

Grades 11-12

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Math Problem Statement

Solution

Part 1: Make the tables

Table 1: Values of g(t)g(t)g(t) for t=48.9,48.99,48.999t = 48.9, 48.99, 48.999t=48.9,48.99,48.999

Table 2: Values of g(t)g(t)g(t) for t=49.1,49.01,49.001t = 49.1, 49.01, 49.001t=49.1,49.01,49.001

Function for g(t)

Values for the tables

Calculate g(t) for each value

Table 1: Values of g(t)g(t)g(t) for t=48.9,48.99,48.999t = 48.9, 48.99, 48.999t=48.9,48.99,48.999

Table 2: Values of g(t)g(t)g(t) for t=49.1,49.01,49.001t = 49.1, 49.01, 49.001t=49.1,49.01,49.001

Part 2: Conjecture about the limit

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Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation

Table 1: Values of $g(t)$ for $t = 48.9, 48.99, 48.999$

Table 2: Values of $g(t)$ for $t = 49.1, 49.01, 49.001$

Table 1: Values of $g(t)$ for $t = 48.9, 48.99, 48.999$

Table 2: Values of $g(t)$ for $t = 49.1, 49.01, 49.001$