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 defined as:
We are asked to find the function defined by:
Step 1: Analyze the behavior as
We are looking at the limit of:
For large , the -terms in both the numerator and denominator dominate. Let’s simplify the expression as .
Step 2: Consider the special case when
When , the expression for becomes:
Thus, for :
Step 3: General result
For , we have:
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$$.Ask a new question for Free
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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