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

$$\sqrt{bemath}$$$$\left(1\right)\underset{x\to \mathrm{\infty }}{\lim }[\frac{x^2}{x+1}-\frac{x^2}{x+3}]?$$$$\left(2\right)provethat{n}^2⩽2^{n}for\mathrm{\forall }n\in \mathbb{N}$$$$bymathematicalinduction$$

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

$$\underset{x\to \mathrm{\infty }}{\lim }\left(\frac{x^3+3x^2-x^3-x^2}{(x+1)(x+3)}\right)=\frac{2x^2}{x^2(1+\frac{1}{x})(1+\frac{3}{x})}=2$$

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