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.