Let's analyze the image. The problem is asking to determine the limits of the following functions and sketch their graphs:

a. $\lim_{x \to 0} (2x + 1)$

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

c. $\lim_{x \to 0} (x^2 + 1)$

d. $\lim_{x \to -1} |x + 1|$

e. $\lim_{x \to 0} \frac{1}{x}$

Solutions:

a. $\lim_{x \to 0} (2x + 1)$

• This is a simple linear function.
• Plugging in $x = 0$: $2(0) + 1 = 1$.
• Answer: $\lim_{x \to 0} (2x + 1) = 1$.

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

• Substitute $x = 2$ directly.
• $\sqrt{2(2) + 1} = \sqrt{4 + 1} = \sqrt{5}$.
• Answer: $\lim_{x \to 2} (\sqrt{2x + 1}) = \sqrt{5}$.

c. $\lim_{x \to 0} (x^2 + 1)$

• Substitute $x = 0$ directly.
• $(0)^2 + 1 = 1$.
• Answer: $\lim_{x \to 0} (x^2 + 1) = 1$.

d. $\lim_{x \to -1} |x + 1|$

• $|x + 1|$ is zero when $x = -1$.
• Answer: $\lim_{x \to -1} |x + 1| = 0$.

e. $\lim_{x \to 0} \frac{1}{x}$

• As $x \to 0$, $\frac{1}{x}$ tends to infinity or negative infinity, depending on the direction.
• This is undefined because of the behavior at $x = 0$ (vertical asymptote).
• Answer: The limit does not exist.

Would you like to see detailed graphs of these functions? Let me know if you have further questions.

Expanding Questions:

1. What happens to the limit $\lim_{x \to 0} (2x + 1)$ if we consider left-hand and right-hand limits?
2. Can you explain why $\frac{1}{x}$ has no limit as $x \to 0$?
3. How does the graph of $|x + 1|$ look around $x = -1$?
4. What is the general shape of the graph for $\sqrt{2x + 1}$?
5. How would the limits change if the points of interest are shifted slightly for each function?

Tip: For piecewise functions like $|x|$, always consider the behavior from both sides of the point of interest to determine the limit.