You've uploaded an image of a set of mathematical problems that involve evaluating limits. Let me extract the relevant text and solve a few of the problems you need help with. One moment while I process the image.The image contains a series of limit problems numbered from 1 to 15. Based on the circled problems (1, 2, 3, 10, 13, and 14), I will solve these first. Let’s break them down step-by-step.

Problem 1:

$\lim_{x \to 1} 2e^{3x + 4}$
Solution:
Substitute $x = 1$:
$2e^{3(1) + 4} = 2e^{7}.$
The limit is: $\boxed{2e^7}.$

Problem 2:

$\lim_{x \to -6} \ln(x + 7)$
Solution:
Substitute $x = -6$:
$\ln(-6 + 7) = \ln(1) = 0.$
The limit is: $\boxed{0}.$

Problem 3:

$\lim_{x \to 2} \log_3(4x + 1)$
Solution:
Substitute $x = 2$:
$\log_3(4(2) + 1) = \log_3(9).$
Since $\log_3(9) = 2$:
$\boxed{2}.$

Problem 10:

$\lim_{x \to \frac{2}{3}} \ln(2x - 4)$
Solution:
Substitute $x = \frac{2}{3}$:
$2\left(\frac{2}{3}\right) - 4 = \frac{4}{3} - 4 = -\frac{8}{3}.$
Since the argument of the natural logarithm is negative, this limit does not exist.
$\boxed{\text{Does not exist}}.$

Problem 13:

$\lim_{x \to 2} \left[\frac{2}{6}(-4x + 3) + 9\right]$
Solution:
Simplify inside the brackets and substitute $x = 2$:
$\frac{2}{6}(-4(2) + 3) + 9 = \frac{2}{6}(-8 + 3) + 9 = \frac{2}{6}(-5) + 9.$
$= -\frac{10}{6} + 9 = -\frac{5}{3} + 9 = \frac{-5 + 27}{3} = \frac{22}{3}.$
The limit is:
$\boxed{\frac{22}{3}}.$

Problem 14:

$\lim_{x \to 5} \left[\log_4(x + 7) - \log_4(x + 1)\right]$
Solution:
Use the logarithmic property $\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)$:
$\log_4(x + 7) - \log_4(x + 1) = \log_4\left(\frac{x + 7}{x + 1}\right).$
Substitute $x = 5$:
$\log_4\left(\frac{5 + 7}{5 + 1}\right) = \log_4\left(\frac{12}{6}\right) = \log_4(2).$
The limit is:
$\boxed{\log_4(2)}.$

Would you like detailed explanations for any of these, or should I solve the other problems as well?

Here are 5 related questions for further practice:

1. How do you solve limits involving exponential functions like $e^{3x+4}$?
2. What happens to a logarithmic limit when the argument approaches zero or becomes negative?
3. How can you simplify expressions like $\log_b(m) - \log_b(n)$?
4. What are common techniques to handle undefined limits like in Problem 10?
5. How does substitution work in evaluating limits?

Tip: Always check the domain of logarithmic and trigonometric functions when evaluating limits!