https://file.aiquickdraw.com/mathbot/file/1727057951906vxv5wzs8.jpg

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.

Tentukan nilai limit berikut dan gambar grafik fungsinya.

Learn how to calculate the limits of functions including linear functions, square root functions, and absolute value functions. This problem covers important calculus concepts such as limits and behavior at infinity. Suitable for high school students in Grades 10-12.

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