The function `y= f(x) = lim_(n to oo) (x^(2n)-1)/(x^(2n)+1)`. Is this function same as the function `g(x) = "sgn"(|x|)-1)`.

Explanation

To determine whether the function \( f(x) = \lim_{n \to \infty} \frac{x^{2n} - 1}{x^{2n} + 1} \) is the same as the function \( g(x) = \text{sgn}(|x|) - 1 \), we will analyze the limit for different cases of \( x \).

### Step 1: Analyze the limit for \( |x| < 1 \)

For \( |x| < 1 \):
\[ f(x) = \lim_{n \to \infty} \frac{x^{2n} - 1}{x^{2n} + 1} \]
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