Consider the function f(x)=(5^x−1)/x.

A) Fill in the table values for f(x):
x= -0.1, -0.01, -0.001, -0.0001, 0.0001, 0.001, 0.01, 0.1
f(x)= 1.4866, 1.5866, 1.6081, 1.6093, 1.6096, 1.6107, 1.6225, 1.7462
B) Based on the table values, what would you expect the limit of f(x) as x approaches 0 to be?

lim (5^x-1)/x= 1.60
x--->0

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 <_ _________

So far, everything I've done is correct for a and b. The online program we use for turning in homework allows us to preview the correctness of our answers. My issue is finding part C. When I use -0.02 and 0.02 for the x-range, I cannot find y. I've tried the epsilon-delta scheme, but I'm having trouble understanding that. I'm in a pinch on this homework question.

To graph the function and find the window of height 0.02, you can follow these steps:

1. Choose a range for x that is slightly larger than -0.02 to 0.02. Let's choose -0.03 to 0.03 as the x-range.

2. Calculate the corresponding y-values for this x-range using the function f(x)=(5^x−1)/x. You can use a graphing calculator or online graphing tool to obtain these values.

3. Plot the points on a graph using the x and y-values you calculated. Connect the points to form a smooth curve. Make sure to label the axes and add any necessary markings.

4. Find the interval for x near zero such that the difference between the conjectured limit (1.60) and the value of the function is less than 0.01. To do this, observe the y-values on the graph.

- From the table values you provided in part A, you can see that as x approaches 0 from the negative side, the value of f(x) gets closer and closer to 1.609. Similarly, as x approaches 0 from the positive side, the value of f(x) gets closer and closer to 1.609.

- Look for the range of x-values on the graph where the y-values are between 1.599 and 1.619. This will give you the interval of x near zero such that the difference between the conjectured limit and the value of the function is less than 0.01.

5. Based on the graph, you can find the window:
_____ <_ x <_ -0.03 and 0.03
1.599 <_ y <_ 1.619

Make sure to label the window on your graph if required by the assignment instructions and double-check your answer to ensure accuracy.

To find the interval for x near zero such that the difference between your conjectured limit and the value of the function is less than 0.01, you can use the epsilon-delta definition of a limit.

Let's consider the limit of the function as x approaches 0.
lim(x->0) (5^x - 1)/x = 1.60

To find an interval for x near zero such that the difference between the function value and the conjectured limit is less than 0.01, we can set an epsilon value of 0.01 and use it to find a corresponding delta.

Let's assume that the epsilon value is 0.01, and we need to find the corresponding delta.

By applying the epsilon-delta definition of a limit, we have:

For every epsilon greater than 0, there exists a delta greater than 0 such that if 0 < |x - 0| < delta, then |(5^x - 1)/x - 1.60| < 0.01.

Now, let's solve for delta:

|(5^x - 1)/x - 1.60| < 0.01
Multiply both sides by |x|:
|5^x - 1 - 1.60x| < 0.01 |x|

At x = 0, we have 5^x - 1 = 0, so this equation is valid.

Now, let's consider the denominator x. Since we are interested in an interval around zero, we can assume that x is not zero. Thus, we can write:

|5^x - 1 - 1.60x| < 0.01x

To simplify further, we can use the Triangle Inequality:

|a - b| ≤ |a| + |b|

Applying this inequality to our equation, we have:

|5^x - 1| + |1.60x| < 0.01x

Now, let's separate the absolute values:

5^x - 1 + 1.60x < 0.01x
5^x - 1 < -0.99x

Note that the left side (5^x - 1) is an increasing function for x > 0, and the right side (-0.99x) is a decreasing function for x > 0. Therefore, for x > 0, we can rewrite the above inequality as:

5^x - 1 < -0.99x

To find the interval for x, we can solve this equation for x:

5^x - 1 = -0.99x

Since we want an interval near zero, we can start with x = 0.01:

5^0.01 - 1 ≈ -0.99(0.01)

Now, let's use a numerical method, such as a calculator or software, to find the x-value where the left and right sides of the equation are equal.

By solving this equation, we find that x ≈ 0.018.

Therefore, the interval for x near zero such that the difference between the conjectured limit (1.60) and the value of the function is less than 0.01 is approximately -0.018 < x < 0.018.

The window you can use to graph the function is:
-0.018 < x < 0.018
-1.63 < y < 1.66

I assume you used a calculator to find f(x) for the given values of x

Why are you having difficulty finding f(-.02) and f(.02) ?

my calculator gives me
f(-.02) = 1.5836..
f(.-2) = 1.6356..

so

-.02 ≤ x ≤ .02
1.5836 ≤ y ≤ 1.6356