It seems you're working on a problem related to limits and approximations. From the image, it looks like you're estimating the value of a limit for the function:

$y = \cos(x) + 2 \sin(3x)$

You're tasked with finding the limit as $x$ approaches 0.5, and estimating values for $\delta$ such that certain conditions are met.

Here's a breakdown of your current inputs:

• L = 2.87 is the limit you’ve estimated as $x$ approaches 0.5.
• The two questions after that ask for values of $\delta$ such that:
  • $|x - 0.5| < \delta$ guarantees $|\cos(x) + 2 \sin(3x) - L| < 0.5$.
  • You also need to estimate for a smaller tolerance, $|\cos(x) + 2 \sin(3x) - L| < 0.02$.

It looks like you're still refining the values for $\delta$ to satisfy those conditions.

Would you like help calculating a more precise value for $\delta$, or do you have any specific questions about the problem?

Here are a few related questions you might find useful:

1. How do you find the limit of a function as $x$ approaches a specific value?
2. What is the definition of the limit of a function in terms of epsilon and delta?
3. How can you interpret the results from a simulation of this kind?
4. What is the difference between absolute error and relative error in approximating limits?
5. How does the behavior of the sine and cosine functions affect the estimation of limits?

Tip: Try using a graphing tool to visualize the function and the behavior of the limit to understand how $\delta$ values change with different tolerances!