The image you uploaded contains eight limit problems. I will go ahead and solve each of them below:

1. $\lim_{x \to 2} (3x + 2)$

We can directly substitute $x = 2$ into the expression: $\lim_{x \to 2} (3x + 2) = 3(2) + 2 = 6 + 2 = 8$ Answer: $\boxed{8}$

2. $\lim_{x \to 2} (x^2 - 1)$

Substituting $x = 2$: $\lim_{x \to 2} (x^2 - 1) = (2^2 - 1) = 4 - 1 = 3$ Answer: $\boxed{3}$

3. $\lim_{x \to 0} \left( 1 + \frac{1}{x} \right)$

As $x \to 0$, the term $\frac{1}{x}$ grows without bound, leading to: $\lim_{x \to 0} \left( 1 + \frac{1}{x} \right) = \infty$ Answer: $\boxed{\infty}$ (undefined as $x$ approaches zero from the right)

4. $\lim_{x \to 5} \sqrt{x - 1}$

Substituting $x = 5$: $\lim_{x \to 5} \sqrt{x - 1} = \sqrt{5 - 1} = \sqrt{4} = 2$ Answer: $\boxed{2}$

5. $\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$

This is an indeterminate form $\frac{0}{0}$, so we factor the numerator: $\frac{x^2 - 1}{x - 1} = \frac{(x - 1)(x + 1)}{x - 1}$ Canceling out the common factor: $\lim_{x \to 1} (x + 1) = 1 + 1 = 2$ Answer: $\boxed{2}$

6. $\lim_{x \to 0} \frac{x^2 - 3x}{x}$

Factoring the numerator: $\frac{x^2 - 3x}{x} = \frac{x(x - 3)}{x}$ Canceling $x$ (for $x \neq 0$): $\lim_{x \to 0} (x - 3) = 0 - 3 = -3$ Answer: $\boxed{-3}$

7. $\lim_{x \to 3} \frac{|x - 3|}{x - 3}$

This limit depends on the direction of $x$ approaching 3:

• As $x \to 3^+$ (from the right), $|x - 3| = x - 3$, so: $\lim_{x \to 3^+} \frac{|x - 3|}{x - 3} = \frac{x - 3}{x - 3} = 1$

• As $x \to 3^-$ (from the left), $|x - 3| = -(x - 3)$, so: $\lim_{x \to 3^-} \frac{|x - 3|}{x - 3} = \frac{-(x - 3)}{x - 3} = -1$

Since the limits from the left and right are not equal, the limit does not exist.

Answer: $\boxed{\text{DNE (Does Not Exist)}}$

8. $\lim_{x \to 0} \frac{|x| - x}{x}$

We evaluate the limit by considering the cases:

• As $x \to 0^+$ (from the right), $|x| = x$, so: $\lim_{x \to 0^+} \frac{|x| - x}{x} = \frac{x - x}{x} = 0$

• As $x \to 0^-$ (from the left), $|x| = -x$, so: $\lim_{x \to 0^-} \frac{|x| - x}{x} = \frac{-x - x}{x} = -2$

Since the limits from the left and right are not equal, the limit does not exist.

Answer: $\boxed{\text{DNE (Does Not Exist)}}$

Would you like further details on any of these problems? I can also draw graphs for the limits if needed.

Here are five additional questions to extend your understanding:

1. What happens to a function's behavior when its limit does not exist at a point?
2. How do piecewise functions affect limit calculations?
3. What are the conditions for a limit to exist at a certain point?
4. How can we use factoring to simplify expressions when calculating limits?
5. What role does one-sided limits play in determining the behavior of functions?

Tip: Always check for indeterminate forms like $\frac{0}{0}$ and $\frac{\infty}{\infty}$, which often require algebraic manipulation or L'Hopital's Rule.