Consider the function f(x)=(4^x−1)/x.

(a) Fill in the following table of values for f(x):

x= -0.1 -0.01 -0.001 -0.0001 0.0001 0.001 0.01 0.1
f(x)=
I the the falues of f(x) for each interval...

(b) Based on your table of values, what would you expect the limit of f(x) as x approaches zero to be?
limx->0 (4^x−1)/x=

(c) Graph the function to see if it is consistent with your answers to parts (a) and (b). By graphing, find an interval for x near zero such that the difference between your conjectured limit and the value of the function is less than 0.01. In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom. What is the window?
______<=x<=______
_______<=y<=______

(a) To find the values of f(x) for each interval, we can simply substitute the given values of x into the function f(x)=(4^x−1)/x and calculate the corresponding values.

For x = -0.1, we have:
f(x) = (4^(-0.1)-1)/(-0.1) = (1/4^0.1 - 1)/(-0.1) ≈ -0.0962

For x = -0.01, we have:
f(x) = (4^(-0.01)-1)/(-0.01) = (1/4^0.01 - 1)/(-0.01) ≈ -0.0096

For x = -0.001, we have:
f(x) = (4^(-0.001)-1)/(-0.001) = (1/4^0.001 - 1)/(-0.001) ≈ -0.00096

For x = -0.0001, we have:
f(x) = (4^(-0.0001)-1)/(-0.0001) = (1/4^0.0001 - 1)/(-0.0001) ≈ -0.000096

For x = 0.0001, we have:
f(x) = (4^(0.0001)-1)/(0.0001) = (4^0.0001 - 1)/(0.0001) ≈ 0.000198

For x = 0.001, we have:
f(x) = (4^(0.001)-1)/(0.001) = (4^0.001 - 1)/(0.001) ≈ 0.002

For x = 0.01, we have:
f(x) = (4^(0.01)-1)/(0.01) = (4^0.01 - 1)/(0.01) ≈ 0.0323

For x = 0.1, we have:
f(x) = (4^(0.1)-1)/(0.1) = (4^0.1 - 1)/(0.1) ≈ 0.672

So, the values of f(x) for each interval are approximately:
f(x) = -0.0962, -0.0096, -0.00096, -0.000096, 0.000198, 0.002, 0.0323, 0.672.

(b) Based on the table of values, as x approaches zero, we can observe that the values of f(x) are getting closer to a certain number, which suggests the existence of a limit. To find the limit of f(x) as x approaches zero, we can take the average of the values from both sides:

limx→0^- f(x) ≈ -0.0962
limx→0^+ f(x) ≈ 0.000198

Therefore, we expect that the limit of f(x) as x approaches zero is approximately 0.

(c) To graph the function and find an interval for x near zero such that the difference between the conjectured limit and the value of the function is less than 0.01, we can use a graphing tool or software. By visually inspecting the graph, we can find such an interval.

The window of height 0.02 can be defined as:
-0.05 <= x <= 0.05
-0.01 <= y <= 0.01

This interval ensures that the graph of the function exits the sides of the window but does not exceed the top or bottom by more than 0.01.

(you can use a graphing tool or software to obtain a visual representation of the function and confirm the window interval)