Prove-using-the-squeeze-law-of-functions-that-lim-x-a-x-a-Recall-that-the-squeeze-theorem-states-that-if-f-x-g-x-h-x-and-lim-x-a-f-x-lim-x-a-h-x-L-then-lim-x-a

Question Number 122192 by physicstutes last updated on 14/Nov/20

$$\mathrm{Prove}\:\mathrm{using}\:\mathrm{the}\:\mathrm{squeeze}\:\mathrm{law}\:\mathrm{of}\:\mathrm{functions}\:\mathrm{that}\: \\ $$$$\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\sqrt{{x}}\:=\:\sqrt{{a}}\:. \\ $$$$\:\mathrm{Recall}\:\mathrm{that}\:\mathrm{the}\:\mathrm{squeeze}\:\mathrm{theorem}\:\mathrm{states}\:\mathrm{that}\:\mathrm{if} \\ $$$$\:{f}\left({x}\right)\:\leqslant\:\mathrm{g}\left({x}\right)\:\leqslant\:{h}\left({x}\right)\: \\ $$$$\:\mathrm{and}\:\underset{{x}−{a}} {\mathrm{lim}}\:{f}\left({x}\right)\:=\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:{h}\left({x}\right)\:=\:{L} \\ $$$$\mathrm{then}\:,\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\mathrm{g}\left({x}\right)\:=\:{L}. \\ $$

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