Math Problem Statement
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.
Solution
Let us solve the problem step by step.
Problem Overview:
We are tasked with finding: where the function is defined as:
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Limits in Complex Analysis
Piecewise Functions
Formulas
Limit Definition: lim(z→a) f(z) = L if f(z) approaches L as z approaches a
Theorems
Continuity and Limit in Piecewise Functions
Suitable Grade Level
Undergraduate (Complex Analysis)
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