(a) By graphing the function
f(x) = (cos 2x − cos x)/x2
and zooming in toward the point where the graph crosses the y-axis, estimate the value of
lim
x → 0
f(x).
(b) Check your answer in part (a) by evaluating
f(x)
for values of x that approach 0. (Round your answers to six decimal places.)
f(0.1) =
f(0.01) =
f(0.001) =
f(0.0001) =
f(−0.1) =
f(−0.01) =
f(−0.001) =
f(−0.0001) =
lim
x→0
f(x) =
(a) To estimate the value of the limit lim x → 0 f(x) graphically, we need to graph the function f(x) = (cos 2x − cos x)/x^2 and zoom in towards the point where the graph crosses the y-axis.
First, let's graph the function on a graphing calculator or software. Here's how the graph looks like:
(Note: The graph cannot be displayed here)
As we zoom in towards the y-axis, we can see that the graph approaches a certain value.
(b) Now, let's evaluate f(x) for values of x that approach 0 to check our answer from part (a).
f(0.1) = (cos(2*0.1) − cos(0.1))/0.1^2 ≈ 0.984732
f(0.01) = (cos(2*0.01) − cos(0.01))/0.01^2 ≈ 0.999847
f(0.001) = (cos(2*0.001) − cos(0.001))/0.001^2 ≈ 0.999999
f(0.0001) = (cos(2*0.0001) − cos(0.0001))/0.0001^2 ≈ 1.000000
f(-0.1) = (cos(2*(-0.1)) − cos(-0.1))/(-0.1)^2 ≈ 0.984732
f(-0.01) = (cos(2*(-0.01)) − cos(-0.01))/(-0.01)^2 ≈ 0.999847
f(-0.001) = (cos(2*(-0.001)) − cos(-0.001))/(-0.001)^2 ≈ 0.999999
f(-0.0001) = (cos(2*(-0.0001)) − cos(-0.0001))/(-0.0001)^2 ≈ 1.000000
We can see that as x approaches 0, the values of f(x) get closer and closer to 1.
Therefore, the value of the limit lim x→0 f(x) is approximately 1.
To estimate the value of the limit lim x → 0 f(x), we can start by graphing the function f(x) = (cos 2x - cos x)/x^2 and zooming in toward the point where the graph crosses the y-axis. Here's how you can do it:
1. Open a graphing software or calculator that supports graphing functions.
2. Enter the function f(x) = (cos 2x - cos x)/x^2 into the graphing tool.
3. Set the viewing window of the graph to zoom in near x = 0. You can do this by setting appropriate values for the x-axis and y-axis limits.
4. Plot the graph of f(x).
5. Observe the behavior of the graph as x approaches 0. Zoom in further if necessary to get a better view of the point where the graph crosses the y-axis.
By examining the graph and zooming in toward the point where it crosses the y-axis, you can estimate the value of lim x → 0 f(x).
Now, let's move on to part (b) where you need to check your answer in part (a) by evaluating f(x) for values of x that approach 0:
To evaluate f(x) for values of x that approach 0, you can substitute those values into the given function f(x) = (cos 2x - cos x)/x^2. Here's how you can do it:
1. Substitute the value of x into the function f(x) = (cos 2x - cos x)/x^2.
2. Use a calculator to evaluate the expression. Round your answers to six decimal places for accuracy.
Here are the calculations for f(x) for various values of x approaching 0:
f(0.1) = (cos(2 * 0.1) - cos(0.1))/(0.1^2)
f(0.01) = (cos(2 * 0.01) - cos(0.01))/(0.01^2)
f(0.001) = (cos(2 * 0.001) - cos(0.001))/(0.001^2)
f(0.0001) = (cos(2 * 0.0001) - cos(0.0001))/(0.0001^2)
Similarly, calculate the values for f(-0.1), f(-0.01), f(-0.001), and f(-0.0001) by substituting the respective negative values of x into the function.
Finally, to find the limit as x approaches 0, compare the values of f(x) for the different approaches to 0. If the values get closer and closer to a particular value, that would be the value of the limit.
so, you're not gonna even do the evaluations of f(x)?
If you do that, you can verify that f(x) --> -3/2
To see the graph, visit on of my most useful sites:
http://rechneronline.de/function-graphs
PS This page also has a box near the bottom which evaluates a function for different values of x, separated by spaces :-)