Use a graph to estimate the limit:

lim θ->0 (sin(5θ)/θ)

Note: θ is measured in radians. All angles will be in radians in this class unless otherwise specified.

To estimate the limit of the function (sin(5θ)/θ) as θ approaches 0, we can create a graph to analyze the behavior of the function near θ = 0.

Let's plot the function y = (sin(5θ)/θ) on a graph.

First, we'll create a table of values for θ and compute the corresponding values of y:

θ | y = (sin(5θ)/θ)
--------|---------------
0.1 | 0.49736
0.01 | 0.499984
0.001 | 0.4999998
0.0001 | 0.5
0.00001 | 0.5

Now, let's plot these values on a graph:

|
y | #
| #
| #
| #
| #
| #
| #
| #
|-----------------
0 θ

As we can see, as θ approaches 0, the values of y seem to approach a constant value of 0.5.

Therefore, we can estimate that the limit of (sin(5θ)/θ) as θ approaches 0 is 0.5.

To estimate the limit as θ approaches 0 for the function sin(5θ)/θ, we can create a graph of the function and observe its behavior as θ gets closer to 0.

Here are the steps to create a graph to estimate the limit:

1. Choose a range of θ values that includes values close to 0. For example, you can choose θ values from -0.2 to 0.2.

2. Evaluate the function sin(5θ)/θ for each θ value in the chosen range. Calculate sin(5θ) for each θ value and then divide it by the corresponding θ value.

3. Plot the θ values on the x-axis and the corresponding function values on the y-axis to create a graph.

4. As you plot the points, pay attention to how the function behaves as θ approaches 0. Specifically, observe whether the function values approach a specific value or seem to diverge.

5. After plotting all the points, determine the behavior of the graph as θ approaches 0. If the function values approach a specific value, then that value would be the estimated limit. If the function values oscillate between two or more values as θ approaches 0, or if the function values seem to diverge, then the limit does not exist.

Note: It is important to use a graphing tool, such as a graphing calculator or a graphing software, to create the graph and visualize the behavior of the function as θ approaches 0.