https://file.aiquickdraw.com/mathbot/file/1725973335461s6ew9j37.jpg

We are given a sequence of functions \( f_n(x) \) defined as:

\[
f_n(x) = \frac{n^2 x^3}{1 + 2n^2 x^2} \quad \text{for every } x \in \mathbb{R}
\]

We are asked to find the function \( f(x) \) defined by:

\[
f(x) = \lim_{n \to \infty} f_n(x)
\]

### Step 1: Analyze the behavior as \( n \to \infty \)

We are looking at the limit of:

\[
f_n(x) = \frac{n^2 x^3}{1 + 2n^2 x^2}
\]

For large \( n \), the \( n^2 \)-terms in both the numerator and denominator dominate. Let’s simplify the expression as \( n \to \infty \).

\[
f_n(x) \approx \frac{n^2 x^3}{2n^2 x^2} = \frac{x^3}{2x^2} = \frac{x}{2}, \quad \text{for } x \neq 0.
\]

### Step 2: Consider the special case when \( x = 0 \)

When \( x = 0 \), the expression for \( f_n(x) \) becomes:

\[
f_n(0) = \frac{n^2 \cdot 0^3}{1 + 2n^2 \cdot 0^2} = 0.
\]

Thus, for \( x = 0 \):

\[
f(0) = 0.
\]

### Step 3: General result

For \( x \neq 0 \), we have:

\[
\lim_{n \to \infty} f_n(x) = \frac{x}{2}.
\]

Thus, the limiting function is:

\[
f(x) =
\begin{cases}
0, & \text{if } x = 0, \\
\frac{x}{2}, & \text{if } x \neq 0.
\end{cases}
\]

Since the provided options are only specific functions, and \( f(x) = \frac{x}{2} \) matches the general form for all real \( x \), the correct answer is:

\[
\boxed{f(x) = \frac{x}{2}}.
\]

---

Would you like further clarification on this, or a deeper breakdown of the steps?

### Related questions:
1. What happens if \( n \) approaches zero instead of infinity?
2. How would the behavior change if the exponent of \( n \) were different in the expression?
3. Can we generalize this limit for other powers of \( x \) and \( n \)?
4. What are the conditions for which limits like this do not exist?
5. How would the limit change if the denominator had different terms?

**Tip**: When analyzing limits of rational functions, focus on the highest powers of \( n \) in both the numerator and denominator to simplify expressions for large \( n \).

For each n = 1, 2, 3,..., define f_n(X) = (n^2 * x^3) / (1 + 2n^2 * x^2) for every x in R. Then the function f(x) defined by f(x) = lim(n -> infinity) f_n(x) exists for each x in R and is equal to one of the following: 1) f(x) = 0, 2) f(x) = x/2, 3) f(x) = x^2, 4) f(x) = x.

Explore how to find the limit of the function sequence f_n(x) = (n^2 * x^3) / (1 + 2n^2 * x^2) as n approaches infinity. This problem involves simplifying rational functions and applying calculus concepts. Suitable for students at an advanced high school or college level.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation