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.

I tried graphing it on my calculator but the limit I got is incorrect.

Soon, if not already, you will come across this limit

limit sinx/x = 1, where x is in radians

your question "almost" looks like that, let's doctor it up so it does

lim sin(5Ø)/Ø as Ø ---> 0
= lim 5 sin(5Ø)/(5Ø) as Ø ---> 0
= lim 5 (1) = 5

When you tried it on your calculator, was it set to radians?
It worked on mine.

Well, it looks like you need a little "graph-ical" assistance! Don't worry, I'm here to help.

Let's take a closer look at the graph of sin(5θ)/θ as θ approaches 0.

Imagine you have a clown juggling five balls. Each ball represents a multiple of θ. As θ gets closer to 0, the balls get smaller and the juggling performance gets more intense.

Now, as θ approaches 0, the balls become so tiny that they seem to disappear. But the show must go on! Our fearless clown keeps juggling air where the balls used to be.

In graph terms, sin(5θ)/θ behaves like a circus performer who juggles an increasingly massive amount of air. While it may seem bizarre, this circus act never ends!

Therefore, the limit of sin(5θ)/θ as θ approaches 0 is equal to... 🥁... 5!

Now, let the clown whoop with joy and continue juggling the air! 🤡🤹‍♀️

To estimate the limit of the function f(θ) = sin(5θ)/θ as θ approaches 0, we can use a graph. However, since you mentioned that your calculator's graph gave an incorrect result, we can also use the properties of limits to find an accurate estimate.

Let's begin by considering the limit as θ approaches 0 from the left side (θ → 0-). We can choose values of θ closer and closer to 0 from the left and observe the corresponding values of f(θ).

θ = -0.1: f(-0.1) = sin(5(-0.1))/(-0.1) ≈ -0.4975/-0.1 ≈ 4.975.
θ = -0.01: f(-0.01) = sin(5(-0.01))/(-0.01) ≈ -0.0248/-0.01 ≈ 2.48.
θ = -0.001: f(-0.001) = sin(5(-0.001))/(-0.001) ≈ -0.000498/-0.001 ≈ 0.498.

As θ gets closer and closer to 0 from the left side, f(θ) approaches 0. It seems that the graph of the function approaches a horizontal line at y = 0 as θ approaches 0 from the left.

Next, let's consider the limit as θ approaches 0 from the right side (θ → 0+). Again, we can choose values of θ closer and closer to 0 from the right and observe the corresponding values of f(θ).

θ = 0.1: f(0.1) = sin(5(0.1))/(0.1) ≈ 0.4975/0.1 ≈ 4.975.
θ = 0.01: f(0.01) = sin(5(0.01))/(0.01) ≈ 0.0248/0.01 ≈ 2.48.
θ = 0.001: f(0.001) = sin(5(0.001))/(0.001) ≈ 0.000498/0.001 ≈ 0.498.

Again, as θ gets closer and closer to 0 from the right side, f(θ) approaches 0. This suggests that the graph of the function approaches a horizontal line at y = 0 as θ approaches 0 from the right.

Since the function approaches the same value (0) from both the left and right sides as θ approaches 0, we can conclude that the limit of f(θ) as θ approaches 0 is 0.

In summary:
lim θ->0 (sin(5θ)/θ) = 0.

To estimate the limit of a function using a graph, you need to plot the values of the function near the value the variable is approaching. In this case, you want to estimate the limit as θ approaches 0.

To graph the function f(θ) = sin(5θ)/θ, you need to choose a set of values for θ that are close to 0. The closer these values are to 0, the better the estimate will be.

Let's start by choosing values of θ close to 0 and calculating the corresponding values of f(θ):

θ | f(θ)
------------------
0.1 | 0.499
0.01 | 0.563
0.001 | 0.563
0.0001| 0.563

By calculating f(θ) for values close to 0, we can observe that the values of f(θ) tend to converge to a specific value, approximately 0.563.

This provides an estimation for the limit as θ approaches 0:

lim θ->0 (sin(5θ)/θ) ≈ 0.563

It is worth noting that this is only an estimation, as the true limit can only be determined mathematically. However, graphing the function and analyzing its behavior for values close to the limit can offer a good approximation.