https://file.aiquickdraw.com/mathbot/file/17266830529213mtnbln6.jpg

The image contains two math problems involving limits. Let me go through them one by one.

1. **First Problem:**

   The first question asks you to evaluate the limit:

   \[
   \lim_{{x \to 2}} (3x + 4)
   \]

   This is a basic limit problem. Since the function \( 3x + 4 \) is a linear and continuous function, you can directly substitute \( x = 2 \) into the expression to find the limit:

   \[
   3(2) + 4 = 6 + 4 = 10
   \]

   So, the solution to the first problem is:

   \[
   \lim_{{x \to 2}} (3x + 4) = 10
   \]

2. **Second Problem:**

   The second part involves a question related to the epsilon-delta definition of a limit:

   It asks, **"If \( \varepsilon = 0.1 \), what should we take \( \delta \) to be so that \( |f(x) - L| < 0.1 \)?"**

   Here, \( f(x) = 3x + 4 \), and from the previous problem, we know that the limit \( L = 10 \). 

   Using the epsilon-delta definition of a limit, we need to find a \( \delta \) such that whenever \( 0 < |x - 2| < \delta \), we ensure that \( |f(x) - L| = |(3x + 4) - 10| < 0.1 \).

   Simplifying the expression \( |f(x) - L| \):

   \[
   |(3x + 4) - 10| = |3x - 6| = 3|x - 2|
   \]

   We need this to be less than 0.1:

   \[
   3|x - 2| < 0.1
   \]

   Dividing both sides by 3:

   \[
   |x - 2| < \frac{0.1}{3} = 0.0333\ldots
   \]

   So, \( \delta = 0.0333 \) would satisfy the condition for \( \varepsilon = 0.1 \).

---

Would you like further clarification on the epsilon-delta process or any other details?

Here are 5 related questions to explore further:

1. What is the epsilon-delta definition of a limit?
2. How would you determine \( \delta \) for a different value of \( \varepsilon \), say \( \varepsilon = 0.01 \)?
3. What does it mean for a function to be continuous at a point?
4. How do you handle limits involving more complex functions (e.g., quadratic or trigonometric)?
5. What is the significance of limits in calculus?

**Tip:** For linear functions like \( f(x) = 3x + 4 \), the epsilon-delta process is straightforward because of the constant slope.

Evaluate the following limit: lim (3x + 4) as x -> 2. If ε = 0.1, what should we take δ to be so that |f(x) - L| < 0.1?

This problem demonstrates how to evaluate the limit of a linear function, 3x + 4, as x approaches 2 using the epsilon-delta definition. It includes step-by-step calculations for finding δ when ε = 0.1. Suitable for high school calculus students.

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