JEE MATHEMATICS MCQ: LIMITS

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\phi(x)=\lim \limits_{{n \rightarrow \infty}} \frac{x^{2 n} f(x)+g(x)}{1+x^{2 n}}

, then

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Correct option: D

\phi(x)= \begin{cases}g(x) \,\text { for }|x| < 1 \\ f(x) \,\text { for }|x| > 1 \\ \frac{f(x)+g(x)}{2} \,\text { for }|x|=1\end{cases}

Option D is correct because it considers the limit of the given function for different ranges of x: |x| < 1, |x| > 1, and |x| = 1.

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