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f-x-1-sin-2x-1-2sin-x-x-g-x-2x-2x-find-lim-x-0-g-f-x-




Question Number 83491 by john santu last updated on 03/Mar/20
f(x) = (((√(1+sin (2x)))−(√(1−2sin (x))))/x)  g(x) = 2x+(√(2x))  find lim_(x→0)  g(f(x))
$$\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{\sqrt{\mathrm{1}+\mathrm{sin}\:\left(\mathrm{2x}\right)}−\sqrt{\mathrm{1}−\mathrm{2sin}\:\left(\mathrm{x}\right)}}{\mathrm{x}} \\ $$$$\mathrm{g}\left(\mathrm{x}\right)\:=\:\mathrm{2x}+\sqrt{\mathrm{2x}} \\ $$$$\mathrm{find}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{g}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\: \\ $$
Commented by john santu last updated on 03/Mar/20
lim_(x→0)  g(f(x)) = g(lim_(x→0)  f(x))  = g(lim_(x→0)  (((√(1+sin (2x)))−(√(1−2sin (x))))/x) )  = g( lim_(x→0)  (((1+((sin (2x))/2))−(1−((2sin (x))/2)))/x))  = g(lim_(x→0)  (((1/2)sin (2x)+sin (x))/x))  = g(2) = 2.2+(√(2.2)) = 4+2 = 6
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{g}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\:=\:\mathrm{g}\left(\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{f}\left(\mathrm{x}\right)\right) \\ $$$$=\:\mathrm{g}\left(\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}+\mathrm{sin}\:\left(\mathrm{2x}\right)}−\sqrt{\mathrm{1}−\mathrm{2sin}\:\left(\mathrm{x}\right)}}{\mathrm{x}}\:\right) \\ $$$$=\:\mathrm{g}\left(\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{1}+\frac{\mathrm{sin}\:\left(\mathrm{2x}\right)}{\mathrm{2}}\right)−\left(\mathrm{1}−\frac{\mathrm{2sin}\:\left(\mathrm{x}\right)}{\mathrm{2}}\right)}{\mathrm{x}}\right) \\ $$$$=\:\mathrm{g}\left(\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\:\left(\mathrm{2x}\right)+\mathrm{sin}\:\left(\mathrm{x}\right)}{\mathrm{x}}\right) \\ $$$$=\:\mathrm{g}\left(\mathrm{2}\right)\:=\:\mathrm{2}.\mathrm{2}+\sqrt{\mathrm{2}.\mathrm{2}}\:=\:\mathrm{4}+\mathrm{2}\:=\:\mathrm{6} \\ $$

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