
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Define \( f_n : \mathbb{R} \to \mathbb{R} \) by
\[ f_n(x) = \frac{\binom{n}{0} + \binom{n}{2} x + \binom{n}{4} x^2 + \ldots}{\binom{n}{1} + \binom{n}{3} x + \binom{n}{5} x^2 + \ldots}, \]
where \( \binom{n}{m} \) is the binomial coefficient. Find a formula for \( f_{n+1}(x) \) in terms of \( f_n(x) \) and \( x \), and determine \( \lim_{n \to \infty} f_n(x) \) for all real \( x \).

Present the answer in LaTex format: \boxed{Your answer}
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