https://file.aiquickdraw.com/mathbot/file/1737217487261fa5p88a3.jpg

Let us solve the problem step by step.

### Problem Overview:
We are tasked with finding:
\[
\lim_{z \to i} f(z)
\]
where the function \( f(z) \) is defined as:
\[
f(z) = 
\begin{cases} 
z^3 + 2z & \text{if } z \neq i, \\
3 + 2i & \text{if } z = i.
\end{cases}
\]

We will calculate the limit using the definition and properties of limits.

---

### Solution:

The value of the limit depends on the continuity of \( f(z) \) at \( z = i \). For the limit to exist, the value of \( \lim_{z \to i} f(z) \) (as \( z \) approaches \( i \)) must match the actual function's definition. 

#### Step 1: Evaluate \( \lim_{z \to i} z^3 + 2z \)
Since the limit considers \( z \neq i \), we use the first part of the function:
\[
f(z) = z^3 + 2z.
\]
Substitute \( z = i \) into \( z^3 + 2z \):
\[
z^3 = i^3 = -i, \quad 2z = 2i.
\]
So:
\[
z^3 + 2z = -i + 2i = i.
\]

#### Step 2: Compare with the second definition
The second part of the definition gives \( f(i) = 3 + 2i \). However, the value of \( \lim_{z \to i} f(z) \) only depends on the first part (since limits concern nearby points, not \( z = i \)).

---

### Final Answer:
\[
\lim_{z \to i} f(z) = i.
\]

---

### Correct Option:
The correct answer is **(b) \( i \)**.

---

Would you like a detailed explanation of why the limit depends on \( z \neq i \) and not the value at \( z = i \)? Here are some related questions for practice:

1. What is the importance of continuity when calculating limits of piecewise functions?
2. If \( f(z) = z^3 + 3z \), what is \( \lim_{z \to 1} f(z) \)?
3. How do we calculate \( \lim_{z \to 0} z^2 + iz \)?
4. Can a function's limit exist even if it is undefined at a point?
5. How does \( \lim_{z \to i} \frac{1}{z - i} \) behave?

---

### Tip:
Always substitute the value of \( z \) into the limit expression after ensuring the function is well-defined in the surrounding region.

If f(z) = {z^3 + 2z, z ≠ i; 3 + 2i, z = i}, evaluate lim f(z) as z approaches i using the definition of limit.

Learn how to calculate the limit of a piecewise function involving complex numbers. This step-by-step solution explains how to evaluate lim f(z) as z approaches i, covering essential concepts in limits and complex functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation