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Question Number 111882 by bemath last updated on 05/Sep/20

\sqrt{b e m a t h}

(1) \lim_{x \to \infty} [\frac{x^{2}}{x + 1} - \frac{x^{2}}{x + 3}] ?

(2) p r o v e t h a t n^{2} ⩽ 2^{n} f o r \forall n \in N

b y m a t h e m a t i c a l i n d u c t i o n

Answered by Dwaipayan Shikari last updated on 05/Sep/20

\lim_{x \to \infty} (\frac{x^{3} + 3 x^{2} - x^{3} - x^{2}}{(x + 1) (x + 3)}) = \frac{2 x^{2}}{x^{2} (1 + \frac{1}{x}) (1 + \frac{3}{x})} = 2