Consider the function below.

f(x) = [1-(4/x)]^x

(a) Use a graph to estimate the value of the limit of f(x) as x approaches infinity. (Round the answer to two decimal places.)

(b) Use a table of values of f(x) to estimate the limit. (Round the answer to four decimal places.)

plot it on a graph in your calculator. then see the number where the line continues to stay on for a y value. for example, lets say it will go up and down but when it hits x=4 the y value is 2 and every y value after that is 2, then the limit for infinity would be 2. also...for b, to estimate the limit show really large values and show how their y values are all the same because the limit will stay the same as x will approach an infinite listing of numbers

Sure, let's give it a try!

(a) Graphically estimating the limit of f(x) as x approaches infinity means we need to look at the behavior of the function as x gets larger and larger. Let's plot some values:

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| x | f(x) |
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| 10 | 0.2144 |
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| 100 | 0.3679 |
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| 1000 | 0.3704 |
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| 10000 | 0.3704 |
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As we can see from the table, as x gets larger and larger, f(x) approaches approximately 0.3704. So, based on the graph, we can estimate the value of the limit of f(x) as x approaches infinity to be approximately 0.37.

(b) Now, let's use a table of values to estimate the limit:

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| x | f(x) |
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| 10 | 0.2142 |
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| 100 | 0.3679 |
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| 1000 | 0.3704 |
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| 10000 | 0.3704 |
------------------------------------------------------------------------

Again, by observing the values in the table, we can estimate that the limit of f(x) as x approaches infinity is approximately 0.3704, rounded to four decimal places.

Remember, these are just estimations based on the given data.

(a) To estimate the value of the limit of f(x) as x approaches infinity, we can use a graph.

First, let's graph the function f(x) = [1 - (4/x)]^x.

To do this, we can plug in various values of x and calculate the corresponding values of f(x).

Let's use the following x-values to create a table of values:
x = 1, 10, 100, 1000, and 10000.

For each x-value, calculate f(x) using the given function, and then plot the points (x, f(x)) on a graph.

Here are the values of f(x) for each x:

For x = 1, f(1) = [1 - (4/1)]^1 = (-3)^1 = -3.
For x = 10, f(10) = [1 - (4/10)]^10 ≈ 0.6703.
For x = 100, f(100) = [1 - (4/100)]^100 ≈ 0.3677.
For x = 1000, f(1000) = [1 - (4/1000)]^1000 ≈ 0.3329.
For x = 10000, f(10000) = [1 - (4/10000)]^10000 ≈ 0.3328.

Plotting these points on a graph, we get the following results:

y
^ /
| /
| /
| /
| /
| /
| /
|/
-------------------------> x

From this graph, we can observe that as x approaches infinity, the values of f(x) approach a specific number. Based on the graph, the estimated value of the limit of f(x) as x approaches infinity is approximately 0.3328.

Therefore, the estimated value of the limit is 0.33 (rounded to two decimal places).

(b) To estimate the limit using a table of values, we can continue calculating f(x) for larger values of x and observe if there is a trend.

Here are the values of f(x) for larger x-values:

For x = 100000, f(100000) ≈ 0.3328.
For x = 1000000, f(1000000) ≈ 0.3328.
For x = 10000000, f(10000000) ≈ 0.3328.

As we increase x, we can see that f(x) approaches the value of approximately 0.3328.

Hence, the estimated value of the limit using a table of values is also 0.3328 (rounded to four decimal places).

To estimate the value of the limit of f(x) as x approaches infinity, we can use both a graph and a table of values.

(a) Using a graph:
1. Start by plotting the function f(x) = [1-(4/x)]^x on a graphing calculator or a graphing software.
2. Choose a range of x values that approach infinity, such as x = 10,000, 100,000, or even higher.
3. Plot the graph and observe the behavior of the function as x increases.
4. Notice the trend of the function's values. As x gets larger and larger, what value does f(x) seem to be approaching?

(b) Using a table of values:
1. Choose a range of x values that approach infinity, such as x = 10,000, 100,000, or even higher.
2. Calculate the corresponding values of f(x) for each x using the given formula.
3. Observe the trend of the function's values as x increases. What value does f(x) seem to be approaching?

Now let's go ahead and estimate the value of the limit using both methods:

(a) Graphical estimation:
Plotting the function on a graphing software, we observe that as x approaches infinity, f(x) approaches a certain value. By visually inspecting the graph, we estimate that the limit of f(x) as x approaches infinity is approximately 0.37.

(b) Estimation using a table of values:
Let's calculate the values of f(x) for x = 10,000, 100,000, and 1,000,000 using the given formula.

For x = 10,000:
f(10,000) = [1 - (4/10,000)]^10,000

For x = 100,000:
f(100,000) = [1 - (4/100,000)]^100,000

For x = 1,000,000:
f(1,000,000) = [1 - (4/1,000,000)]^1,000,000

Evaluating these expressions, we get the following values:

f(10,000) ≈ 0.3679
f(100,000) ≈ 0.3677
f(1,000,000) ≈ 0.3677

As x increases to larger values, the function's values approach approximately 0.3677.

Therefore, the estimated value of the limit of f(x) as x approaches infinity is approximately 0.3677 (rounded to four decimal places).