Math Problem Statement

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.

Solution

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

fn(x)=n2x31+2n2x2for every xRf_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)f(x) defined by:

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

Step 1: Analyze the behavior as nn \to \infty

We are looking at the limit of:

fn(x)=n2x31+2n2x2f_n(x) = \frac{n^2 x^3}{1 + 2n^2 x^2}

For large nn, the n2n^2-terms in both the numerator and denominator dominate. Let’s simplify the expression as nn \to \infty.

fn(x)n2x32n2x2=x32x2=x2,for x0.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=0x = 0

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

fn(0)=n2031+2n202=0.f_n(0) = \frac{n^2 \cdot 0^3}{1 + 2n^2 \cdot 0^2} = 0.

Thus, for x=0x = 0:

f(0)=0.f(0) = 0.

Step 3: General result

For x0x \neq 0, we have:

limnfn(x)=x2.\lim_{n \to \infty} f_n(x) = \frac{x}{2}.

Thus, the limiting function is:

\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$$.

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Math Problem Analysis

Mathematical Concepts

Calculus
Limits
Sequences of functions
Rational functions

Formulas

f_n(x) = (n^2 * x^3) / (1 + 2n^2 * x^2)
f(x) = lim(n -> infinity) f_n(x)

Theorems

Limit of a function sequence
Simplifying rational functions

Suitable Grade Level

Grades 11-12, College level